Chapter 1

Theory of Vector Optimization

Alfred Göpfert and Christiane Tammer Department of Mathematics and Computer Science Martin-Luther-University Halle-Wittenberg D-06099 Halle, Germany goepfert@mathematik.uni-halle.de, tammer@mathematik.uni-halle.de

Abstract: We introduce several solution concepts for multicriteria optimization problems, give a characterization of approximately efficient elements and discuss a general scalarization procedure. Furthermore, we derive necessary and sufficient optimality conditions, a minimal point theorem, a vector-valued variational principle of Ekeland’s type, Lagrangean multiplier rules and duality statements. An overview on vector variational inequalities and vector equilibria is given. Moreover, we discuss the results for special classes of vector optimization problems (vector-valued location and approximation problems, multicriteria fractional programming and optimal control problems).

1.1 Solution concepts

Keywords: Vector optimization, approximately efficient elements, scalarization.

1.1.1 Minimality notions in partially ordered spaces

Many practically important problems can be described by a multicriteria optimization problem with more than one objective function. In order to introduce a solution concept for such problems we consider a nonempty subset M of a linear topological space Y and a reflexive, transitive and antisymmetric relation R on Y (and so also on M) which gives an order structure (a partial order) on Y (and M). If two elements m_{1}, m_{2} of M are comparable with respect to R we write (m_{1}, m_{2}) ∈ R or m_{1} ≤ m_{2}. With help of such an order relation one is able to define what a minimal element m^{0} ∈ M is:

∄m ∈ M \ {m^{0}} with m ≤ m^{0}.

Normally, M ⊂ Y is a set which is given (in any way) by the decision maker. But how to choose really a preference relation R is in a wide sense an interesting question and touches also questions of sensitivity of the problem dealt with. A vector minimization problem or vector optimization problem (v.o.p.) is a problem to determine efficient elements of a certain set in the following sense:

Definition 1.1.1 (Efficiency): Given Y, M, R as above, minimal elements of M with respect to R are called efficient and the set (possibly empty) of all efficient elements on M with respect to R is denoted by Min(M, R). It is often the case that a relation R_{C} on Y is generated by a proper convex pointed cone C ⊂ Y, then R_{C} := {(y_{1}, y_{2}) | y_{2} − y_{1} ∈ C} is antisymmetric and we write Eff(M, C) for Min(M, R_{C}). For a v.o.p. with an objective function f : U ⟶ Y and a feasible set U we write:

(P) Determine the set Eff(f[U], C),

or sometimes simply f(x) ⟶ v − min, x ∈ U, suppressing the order defining cone C.

Throughout this chapter Y is a linear topological space, Y^{∗} is its topological dual, C ⊂ Y is a convex cone, i.e., C + C ⊂ C and [0, ∞[C ⊂ C, C^{∗} = {y^{∗} ∈ Y^{∗} | y^{∗}(y) ≥ 0 ∀y ∈ C} is the dual cone of C and C^{∗0} = {y^{∗} ∈ Y^{∗} | y^{∗}(y) > 0 ∀y ∈ C \ {0}}. Furthermore, we assume that C is pointed, i.e., C ∩ −C = {0}. We denote for a subset B of a linear topological space Y the topological interior of the set B by int B, the topological closure of B by cl B and the topological boundary of B by bd B. A functional z : Y ⟶ R is called B-monotone if y^{1} ∈ y^{2} + B implies z(y^{1}) ≥ z(y^{2}) for all y^{1}, y^{2} ∈ Y. We say that a B-monotone functional z is even strictly B-monotone if, additionally, y^{1} ∈ y^{2} + (B \ {0}) implies z(y^{1}) > z(y^{2}).

The importance of the mentioned general efficiency concept for decision making was pointed out by Yu [300] for convex cones in R^{n} (compare also [11], [195], [284], [285]). Yu’s concept has been extended by various authors to more general spaces (see [136], [137], [138]), and by Gerstewitz and Iwanow [106] and by Weidner [280] for general sets D. Several efficiency concepts in stochastic multiple objective programming are given in [41]. For definitions on approximate efficiency see Definition 1.1.3.

Definition 1.1.2 (Weak, proper and strongly proper efficiency): Let Y, M, C as above.

(i) Assume int C is nonempty and y^{0} ∈ M. If y^{0} ∈ Eff(M, {0} ∪ int C), then y^{0} is called weakly efficient and we write y^{0} ∈ w − Eff(M, C).

(ii) Assume that there is a cone H ⊂ Y, H ≠ Y, H ≠ C such that C \ {0} ⊂ int H and y^{0} ∈ M. If y^{0} ∈ Eff(M, H) then y^{0} is called properly efficient on M w.r. to C and H and we write y^{0} ∈ p − Eff(M, C) (having H in mind).

(iii) Assume that D is a proper convex cone in Y and U_{Y} a fundamental system of neighbourhoods of zero in Y such that the pair (C, D) has the property

∀W ∈ U_{Y} ∃V ∈ U_{Y} : (C \ W) + V ⊂ D,  (1.1.1)

then y^{0} ∈ M is called strongly proper efficient on M w.r.to C, if y^{0} ∈ Eff(M, D). We write y^{0} ∈ SP Eff(M, C).

Dependant on H and D there are a lot of special proper-efficiency concepts in the literature (cf. [27], [103], [106], [133], [145], [150], [171], [235], [303]).

Obviously

SP Eff(M, C) ⊂ p − Eff(M, C) ⊂ Eff(M, C) ⊂ w − Eff(M, C) (1.1.2)

and this chain of inclusions, giving in some sense a set-valued estimation of the wanted set Eff(M, C), justifies Definition 1.1.2. Further results (generalizations of the theorem of Arrow, Barankin and Blackwell) are given by Jahn [150] and Ferro [97]. But much more can be proved, so w − Eff(M, C) often possesses properties like connectedness (cf. Luc [190]) or closedness (whereas the efficiency set does not), and p − Eff(M, C) permits dependant on the choose of H an interpretation by scalarization, which can be used for real calculations of elements of Eff(M, C), compare Section 1.1.2. Generally, the last two inclusions in (1.1.2) are not equalities as can be seen by the following example.

Example 1.1.1 Assume Y = R^{2}, M = {A_{1} = (0, 1), A_{2} = (0, 0), A_{3} = (1, −1)} ⊂ Y, C = R_{+}^{2}, H = {(y_{1}, y_{2}) ∈ Y : y_{2} > 0} ∪ {(y_{1}, 0) ∈ Y : y_{1} ≥ 0}. Then C and H are pointed cones and

Eff(M, C) = {A_{2}, A_{3}},
w − Eff(M, C) = M = Eff(M, 0 ∪ int C),
Eff(M, H) = {A_{3}} ∈ p − Eff(M, C).

A geometrical characterization of weakly efficient elements is given by Carrizosa and Plastria [50]. Existence results for efficient elements are shown by Borwein ([31], [32]), Cesari and Suryanarayana ([52]), Chew [66], Corley [70], Dedieu [77], Dolecki and Malivert [84], Gajek and Zagrodny [102], [101], Ha [127], [128], Hartley [130], Hazen and Morrin [131], Henig [132], Isac [136], [137], [138], Jahn [147], Krasnosel’skij [164], Luc [189], Postolica [223], [224], Sonntag and Zalinescu [246], Sterna-Karwat [250], [251], Takahashi [252]. Furthermore, stability results for setvalued mappings and vector optimization problems are presented by [9], [13], [14], [15], [84], [95], [96], [135], [160], [166], [167], [191], [205], [206], [213], [214], [216], [217], [218], [238], [239], [243], [248], [249], [258], [268], [270], [304], [305].

1.1.2 Scalarization Methods and Separation Theorems

The following separation theorems given in [110] play an important role in the theory of multicriteria optimization since they permit (under conditions) scalarizing of vector optimization problems (compare [105], [106], [110], [153], [190], [211], [235], [281]). We give a first separation theorem in linear topological spaces without any convexity assumptions (see [110]).

Theorem 1.1.1 Let us assume that:

(i) Y is a linear topological space;

(ii) C ⊂ Y is a cone with int C ≠ ∅;

(iii) D is a proper subset of Y with non-empty interior such that cl D + int C ⊂ int D;

(iv) A ⊂ Y is a non-empty set of Y.

Then the following statements are true:

(a) A ∩ (−int D) = ∅ implies that there exists a continuous functional z : Y ⟶ R, which is strictly (int C)-monotone with the range (−∞, +∞) and

z(A) ≥ 0, z(−cl D) ≤ 0,
z(−bd D) = 0, z(−int D) < 0,
and z(int A) > 0 if int A ≠ ∅.

(b) If cl D is a convex set, then the functional z in (a) can be chosen such that it is also convex.

(c) One can construct the functional z in (a) such that z is B-monotone for each set B ⊂ Y with bd D + B ⊂ cl D and strictly B- monotone, if bd D + (B \ {0}) ⊂ int D.

(d) If bd D + bd D ⊂ cl D, then z in (a) can be chosen such that it is subadditive on Y.

In the proof of this separation theorem we use a functional z : Y ⟶ R introduced by Gerstewitz [105] (compare also Gerth and Weidner [110]) which is defined for an arbitrary set D ⊂ Y given by (iii) with int D ≠ ∅, cl D + int C ⊂ int D and for a fixed vector k^{0} ∈ int D in the following way:

z(y) := inf{t ∈ R | y ∈ tk^{0} − cl D},  y ∈ Y.  (1.1.3)

This functional has the following nice properties (cf. [110]):

z(y) < r ⟺ y ∈ −int D + rk^{0},  (1.1.4)
z(y) ≤ r ⟺ y ∈ −cl D + rk^{0},  (1.1.5)
z(y) = r ⟺ y ∈ −bd D + rk^{0},  (1.1.6)
z(y) ≥ r ⟺ y ∉ −int D + rk^{0},  (1.1.7)
z(y) > r ⟺ y ∉ −cl D + rk^{0}.  (1.1.8)

From Theorem 1.1.1 we derive the following corollary.

Corollary 1.1.1 Assume that:

(i) Y is a linear topological space; (ii) C ⊂ Y is a proper convex cone with non-empty interior;

(iii) A ⊂ Y is a proper nonempty subset of Y.

Then A ∩ −int C = ∅ implies that there exists a continuous sublinear functional z : Y ⟶ R which is strictly int C- monotone with the range (−∞, +∞) and

z(A) ≥ 0, z(−cl C) ≤ 0, z(−bd C) = 0,
z(−int C) < 0, z(C) ≥ 0, z(int C) > 0
and z(int A) > 0 if int A ≠ ∅.

In Theorem 1.1.1 and Corollary 1.1.1 we deal with strictly (int C)- monotone functionals. If C ≠ Y is a convex cone with non-empty interior, then the strict (int C)-monotonicity implies the C-monotonicity.

In Theorem 1.1.1 we have supposed that C is a cone with non-empty interior. This property is not guaranteed in many important cases; for example, even the usual ordering cone L_{2}^{+} := {x ∈ L_{2}[0, 1] : x(t) ≥ 0 ∀t ∈ [0, 1] almost everywhere} in L_{2}[0, 1] has no interior point. Therefore, we formulate a separation theorem in which the assumptions on D do not depend on a cone C with non-empty interior. In this case we have to demand the directedness of Y (this is (iv) in the following theorem) with respect to the closure of D explicitly.

Theorem 1.1.2 Second Separation Theorem. Let us assume:

(i) Y is a linear topological space;

(ii) D is a proper convex and open subset of Y;

(iii) there exists an element k^{0} ∈ Y such that cl D + αk^{0} ⊂ cl D for each α ≥ 0;

(iv) Y = ∪{−cl D + αk^{0} | α ∈ R};

(v) A ⊂ Y is a non-empty subset of Y.

Then the following statements are true:

(a) A∩(−D) = ∅ implies that there exists a continuous convex functional z : Y ⟶ R with the range (−∞, +∞) and

z(A) ≥ 0, z(−bd D) = 0, z(−D) < 0,
and z(int A) > 0 if int A ≠ ∅.

(b) One can construct the functional z in (a) such that z is B-monotone for each set B ⊂ Y with bd D + B ⊂ cl D.

(c) If B ⊂ Y and bd D + (B \ {0}) ⊂ D then one can construct the functional in (a) such that it is also strictly B- monotone.

(d) If bd D + bd D ⊂ cl D, then z in (a) can be chosen such that it is subadditive on Y.

The separation theorems generalize assertions by Gerstewitz and Iwanow [105] for a not necessary convex set A and a convex set D which were proved in another way by using stronger assumptions.

Example 1.1.2 Choose

Y = R^{2}, k^{0} = (1, 1)^{T} ,
D := {(−y_{1}, −y_{2})^{T} | y_{1} + y_{2} < 1}}
∪ {(−y_{1}, −y_{2})^{T} | ∃λ ∈ R : ||(y_{1}, y_{2})^{T} − λk^{0}||_{2} < 1}.

Y, k^{0} and D meet all the assumptions of Theorem 1.1.2 except the convexity of D. The functional z defined by (1.1.3) has no finite value in k^{0} ∈ −D.

Example 1.1.3 Choose Y, k^{0} as in the example above and

D := {(y_{1}, y_{2})^{T} | ∃λ ∈ R : ||(y_{1}, y_{2})^{T} − λk^{0}||_{2} < 1}.

Y, k^{0} and D meet all conditions in Theorem 1.1.2 except

Y = ∪{−cl D + αk^{0} | α ∈ R}.

The functional z constructed in (1.1.3) takes no finite value in k^{0} ∈ D; moreover, it attains nowhere finite values on D.

Scalarization of a given v.o.p. means converting of that problem into an optimization problem (or a family of such ones) with a common real valued function to minimize (cf. [110], [144], [149], [146], [190]). If solutions of the latter problems (often called scalarized problems) are also solutions of the given v.o.p., then the made scalarization seems to be advantageous in order to solve the v.o.p. since methods of common ”scalar” optimization (nonlinear programming) can be used. Mostly advantageous scalarizing is done by using suitable monotone functionals, where the following propositions serve as theoretical background.

Proposition 1.1.1 Let Y be a linear space partially ordered by a nontrivial pointed convex cone C, M a nonvoid set in Y and s : M ⟶ R a functional with dom s = M. If for some y^{0} ∈ M

s(y) ≥ s(y^{0}) ∀y ∈ M  (1.1.9)

then y^{0} ∈ Eff(M, C) if one of the following conditions is fulfilled:

(i) s is monotoneously increasing on M and y^{0} is uniquely determined,

(ii) s is strictly monotone increasing on M.

Proof: We consider y^{0} ∈ M such that (1.1.9) is valid. If y^{0} ∉ Eff(M, C) then there exists y^{1} ∈ (y^{0} − (C \ {0})) ∩ M, y^{1} ≠ y^{0}. So −(y^{1} − y^{0}) ∈ C \ {0}, and if s is monotoneously increasing, then s(y^{0}) ≥ s(y^{1}), which together with (1.1.9) gives s(y^{1}) = s(y^{0}), y^{1} ≠ y^{0}, in contradiction to (i). If s is strictly monotone we get s(y^{1}) < s(y^{0}), a contradiction too.

So Proposition 1.1.1 seems especially to be advantageous if one has strictly monotone functionals. In particular this is the case if Y is a Banach space and C^{∗0} ≠ ∅, obviously. Then efficiency can even be characterized by scalarization:

Proposition 1.1.2 Let Y be a Banach space, partially ordered by a nontrivial convex pointed cone C ⊂ Y with C^{∗0} ≠ ∅ and ∅ ≠ M ⊂ Y. y^{0} ∈ Eff(M, C) iff there are c ∈ Y and y^{∗} ∈ C^{∗0} such that y^{0} solves the scalarized optimization problem

y^{∗}(y) ⟶ min s.t. y ∈ M, y ≤_{C} c.  (1.1.10)

Proof: Let y^{0} be a solution of (1.1.10), then y^{0} ∈ Eff(M, C). Otherwise there exists ȳ ∈ M such that ȳ ∈ y^{0} − (C \ {0}). So y^{∗} ∈ C^{∗0} gives y^{∗}(ȳ) < y^{∗}(y^{0}) in contradiction to 1.1.10 since ȳ ∈ y^{0} − C ∈ c − C − C = c − C, i.e. ȳ ≤ c. If y^{0} ∈ Eff(M, C) put c := y^{0}. Take y^{∗} ∈ C^{∗0}, then y^{0} solves 1.1.10 with the chosen c. Indeed, if there were a feasible ȳ such that y^{∗}(ȳ) < y^{∗}(y^{0}) it follows ȳ ≠ y^{0} since y^{∗} ∈ C^{∗0}. But also ȳ ≤_{C} c = y^{0} and so y^{0} ∉ Eff(M, C), a contradiction.

We assume for the rest of this section

(i) Y is a linear topological space;

(ii) C is a cone in Y with nonempty interior; k^{0} ∈ int C;

(iii) D is a proper subset of Y with nonempty interior such that cl D + int C ⊂ int D.

The properties of the functional z defined by (1.1.3) can easily be used in order to characterize weak efficiency. Theorem 1.1.3 is an example of such a result. Furthermore, if we suppose additional assumptions for the set D then additional properties follow for z. Theorem 1.1.3 and 1.1.4 both collect such results.

Theorem 1.1.3 Suppose additionally to the assumptions given above

(A_{1}) Let D ⊂ Y be a set with int D ≠ ∅ and 0 ∈ D \ int D for which there exists a cone C with nonempty interior such that cl D + int C ⊂ int D.

Then y^{0} ∈ w − Eff(M, D) if and only if y^{0} ∈ M and there exists a continuous functional z : Y ⟶ R which is strictly (int C)-monotone (even strictly int Dmonotone if D is a convex cone) with the range (−∞, +∞) and with the properties

z(y^{0}) = 0,  z(M) ≥ 0,  z(int M) > 0,  z(y^{0} − bd D) = 0 (1.1.11)
z(y^{0} − D) ≤ 0,  z(y^{0} − int D) < 0.  (1.1.12)

If cl D is convex, z can be chosen convex.

(A_{2}) If int D is convex, 0 ∈ cl (int D) \ int D and if there is k^{0} ∈ Y with

cl (int D) + αk^{0} ⊂ cl (int D), for each α ≥ 0,

and

Y = ∪{cl (int D) + αk^{0} | α ∈ R},

then y^{0} ∈ w − Eff(M, D) if and only if y^{0} ∈ M and there exists a continuous convex functional z : Y ⟶ R with the range (−∞, +∞) such that (1.1.11) and (1.1.12) hold.

Theorem 1.1.4 Let Y, M, D, y^{0} be as in Theorem 1.1.3. If condition (A_{1}) (or (A_{2})) holds, we have for z

(a) z can be chosen (int D)-monotone, if there holds bd D + int D ⊂ cl D (or bd (int D) + int D ⊂ cl (int D)).

(b) z can be chosen D-monotone, if there holds bd D + D ⊂ cl D (or bd (int D) + D ⊂ cl (int D)).

(c) z can be chosen strictly (int D)-monotone, if bd D + int D ⊂ int D (or bd (int D) + int D ⊂ (int D)).

(d) z can be chosen subadditive, if there holds bd D + bd D ⊂ y^{0} + cl D (or bd (int D) + bd (int D) ⊂ y^{0} + cl (int D)).

In some sense the following Theorem 1.1.5 is converse to Theorem 1.1.4. Furthermore, the following theorem gives a characterization of properly efficient elements.

Theorem 1.1.5 : Assume D as in Theorem 1.1.3 and z a strictly D-monotone functional and let y^{0} ∈ M be a point in which z attains its minimum on M. Then, these results hold:

(a) y^{0} is an efficient element of M with respect to D.

(b) If z is continuous, there exists an open set H ⊃ (D\{0}), with cl H+(D\{0}) ⊂ H and y^{0} ∈ Eff(M, H).

(c) if z is convex, there exists a convex set H ⊃ (D\{0}), such that y^{0} ∈ Eff(M, H).

(d) If z is continuous and convex, there exists an open convex set H ⊃ (D \ {0}) with cl H + (D \ {0}) ⊂ H, such that y^{0} ∈ Eff(M, H).

(e) If 0 ∈ cl (D\{0}) holds, H in (b), (c), (d) can be chosen such that 0 ∈ cl H \H.

(f) If z is linear, there exists a set H ⊃ (D \ {0}) such that H ∪ {0} is a convex cone and y^{0} ∈ Eff(M, H).

(g) If z is linear and continuous, H in (b) can be chosen such that H ∪ {0} is a convex cone.

Example 1.1.4 If 0 ∉ cl (D \ {0}), then the set H constructed in the proof of Theorem 1.1.5 can fail to fulfil 0 ∈ cl H. For example, take

Y = R,  D = R_{+} \ (0, 1),  M = [1, 4],

and define

z(y) = \{ \begin{matrix}y + \frac{1}{2} : & y < \frac{1}{2}, \\ 1 : & \frac{1}{2} ≤ y ≤ 1, \\ y : & y > 1. \end{matrix}

z is continuous and strictly D-monotone and attains its minimum on F in y^{0} := 1. Note that 0 is not a cluster point of the set

H = \{y ∈ Y | z(−y + y^{0}) < z(y^{0})\} = (\frac{1}{2}, ∞).

Considering properly efficient points instead of weakly efficient ones one can prove connections to scalarizing functionals z similar as in the theorems above. As an example we give

Theorem 1.1.6 Let be y^{0} ∈ Eff(M, H) for an open convex set H ⊃ (D \ {0}) such that 0 ∈ cl H \ H. Assume either condition (A_{1}): There exists an element k^{0} ∈ Y with cl H + αk^{0} ⊂ cl H for each α ∈ R_{+} and Y = ∪{cl H + αk^{0} | α ∈ R} or condition (A_{2}): Let D be a cone with nonempty interior and cl H + int D ⊂ H. Then

(a) There exists a continuous functional z : Y ⟶ R with the range (−∞, +∞) such that (1.1.11) and (with H instead of D) (1.1.12) hold. Under (A_{1}) z can be chosen convex, under (A_{2}) strictly (int D)- monotone. In the last case convexity of z follows if cl H is convex.

(b) If bd H + D ⊂ cl H, z can be chosen such that it is D-monotone.

(c) If bd H + (D \ {0}) ⊂ H, z can be chosen such that it is strictly D-monotone.

(e) If bd H + bd H ⊂ y^{0} + cl H, z can be chosen such that it is subadditive.

Theorems 1.1.1 - 1.1.6 in full generality stem from a paper by Gerth and Weidner [110]. But of course special cases of those theorems had been proved earlier, f.e., scalarization results for weakly efficient points for the case Y = R^{n} and B = y^{0}−R_{+}^{n}, k^{0} ∈ int R_{+}^{n} with the functional

z(y) = inf {t ∈ R | y ∈ y^{0} − R_{+}^{n} + tk^{0}},

had been obtained for k^{0} = (1, ..., 1)^{T} by Brosowski and Conci [39].

1.1.3 Approximate minimality

In the last years several concepts for approximately efficient solutions of a vector optimization problem were published (compare [79], [80], [118], [120], [119], [134], [139], [159], [185], [201], [202], [208], [247], [253], [254], [258], [274]). The reason for introducing approximately efficient solutions is the fact that numerical algorithms usually generate only approximative solutions anyhow and moreover, the efficient point set may be empty, whereas approximately efficient points always exist under very weak assumptions.

To find an access to approximate efficiency we will use our concept given in [108] by means of nonempty subsets M and B of a linear topological space Y, a proper, convex pointed cone C ⊂ Y , C ≠ Y, with int C ≠ ∅, cl B + int C ⊂ int B, k^{0} ∈ int C and a real number ϵ ≥ 0. Relations to some other concepts follow afterwards.

Definition 1.1.3 (Approximate efficiency, ϵk^{0}- efficiency) An element y_{ϵ} ∈ M is said to be ϵk^{0}-efficient on M with respect to B as above if

M ∩ (y_{ϵ} − ϵk^{0} − (B \ {0})) = ∅.

We denote the set of ϵk^{0}-efficient points of M with respect to B by Eff(M, B_{ϵk^{0}}). For the case ϵ = 0 and B = C the set Eff(M, B_{ϵk^{0}}) coincides with the usual set Eff(M, C) of efficient points of M with respect to C as given in Definition 1.1.1. Obviously for ϵk^{0} + (B \ {0}) ⊂ B \ {0} it holds

Eff(M, B) ⊆ Eff(M, B_{ϵk^{0}})

for any ϵ ≥ 0 and k^{0} ∈ int C.

Approximate efficiency can also be defined by scalarization:

Definition 1.1.4 : Let z : Y ⟶ R be any C-monotone functional. An element y_{ϵ} ∈ M is said to be ϵ-efficient with respect to z if

y ∈ y_{ϵ} − C ⟹ z(y_{ϵ}) ≤ z(y) + ϵ

holds for each y ∈ M. We denote the set of such ϵ-efficient points of M with respect to z by ϵ − Eff^{z}(M).

Simple examples show that generally ϵ − Eff^{z}(M) ≠ Eff(M, B_{ϵk^{0}}). So it seems to be very interesting to study relations between the Definitions 1.1.3 and 1.1.4 (cf. [258]). We use the functional z : Y ⟶ R introduced in Section 1.1.2 by (1.1.3).

Theorem 1.1.7 Let B ⊂ Y with int B ≠ ∅, C ⊂ Y a cone with int C ≠ ∅, cl B + int C ⊆ int B, bd B + bd B ⊂ cl B, 0 ∈ bd B, k^{0} ∈ int C and ϵ > 0. Then z given in (1.1.3) is a continuous, strictly (int C)-monotone functional with the range (−∞, ∞) and for any y_{ϵ} ∈ Eff(M, B_{ϵk^{0}}) and y ∈ M we have

ẑ(y_{ϵ}) ≤ ẑ(y) + ϵ

for ẑ(y) := z(y − y_{ϵ}), where ẑ is again a continuous, strictly (int C)-monotone functional with the range (−∞, ∞).

Additionally, if cl B + (C \ {0}) ⊂ int B, then the functional z (and hence ẑ also) is strictly C-monotone.

Theorem 1.1.8 Let us consider a cone C ⊂ Y with int C ≠ ∅, a set B ⊂ Y with int B ≠ ∅, cl B + int C ⊆ int B, an element k^{0} ∈ int C, ϵ ≥ 0 and assume that z given in (1.1.3) is strictly C-monotone, subadditive and continuous on Y. If y_{ϵ} ∈ M fulfils the inequality

z(y_{ϵ}) ≤ z(y) + ϵ  (1.1.13)

for each y ∈ M, then there is an open set D ⊂ Y with C \ {0} ⊂ D, 0 ∈ cl D \ D and cl D + (C \ {0}) ⊂ D such that

y_{ϵ} ∈ Eff(M, D_{ϵk^{0}}).

Corollary 1.1.2 :

1. Under the assumptions of Theorem 1.1.7 it holds:

y_{ϵ} ∈ Eff(M, B_{ϵk^{0}}) ⟹ y_{ϵ} ∈ ϵ − Eff ẑ(M)

with ẑ(y) = z(y − y_{ϵ}) and z from (1.1.3).

2. Under the assumptions of Theorem 1.1.8 it holds:

y_{ϵ} ∈ ϵ − Eff^{z}(M) ⟹ y_{ϵ} ∈ Eff(M, D_{ϵk^{0}}),

with z given by (1.1.3).

1.1.4 Conclusions

In the last years several concepts of approximately efficient elements have been introduced, followed, recently, by characterizations of approximately efficient elements and necessary and sufficient conditions for such elements. In order to describe solution procedures for multicriteria optimization problems it is important to discuss useful scalarization procedures, especially monotony properties of scalarizing functionals (or utility functions). Of course, studies of dependence of solutions if data of the given problem vary, are always basic tasks.

1.2 Optimality Conditions

Keywords: Minimal point theorem, variational principle, Lagrangean multipliers, saddle point assertions.

1.2.1 Maximal Point Theorems and Variational Principles

Dealing with extremal problems one of the main objectives is to derive optimality conditions. Therefore very often variational methods are used skilfully linked with the introduction of a perturbation of the problem. Important results of duality, optimality and saddle point theory have been obtained in this way. Since the seventies with Ekeland’s variational principle and some equivalent results an essentially new approach was available for deriving optimality conditions.

Ekeland’s variational principle [89] is a momentous assertion about the existence of an exact solution of a slightly perturbed optimization problem in a neighbourhood of an approximate solution of the original problem. Many authors have published extensions and applications of Ekeland’s variational principle as well as equivalent statements (cf. Brezis and Browder [37], Brondsted [38], Rockafellar [229], Oettli [208], Penot [215], Daneš [75], Borwein and Preiss [33], Isac [139], Rolewicz [230], Georgiev [104], De Figueiredo [78], Takahashi [252], Phelps [221], Attouch and Riahi [9], Gajek and Zagrodny [102], Göpfert, Tammer and Zalinescu [121], [122]).

In the background of the mentioned variational principle there exist conical support points of arbitrary closed sets in the considered spaces. Of course, conical support points are closely related with supporting of convex sets by halfspaces, being the base of convex duality as well as the background of a lot of optimality conditions in convex analysis.

The results of Bishop and Phelps [21] from 1962 concerning supporting points of (convex) sets can also be seen as a first contribution to the topic of Ekeland’s variational principle, given in the form of a maximal point theorem.

Extremal problems do not only exist for goal functionals having R^{1} as image space but a more general halfordered space, where chiefly the order is given by a cone in this space. Such problems are the so called multiobjective extremal problems. Also for such problems (and even more general ones, think of set valued or stochastic functions which may be involved in extremal problems) optimality conditions have been derived.

Loridan [185] in 1984 was the first, who successfully made use of a special multiobjective variational principle, discovered by himself, to get optimality conditions. He [185] has presented a vector-valued variational principle for the finite-dimensional case using a scalarization and Ekeland’s original result. Further, Chen and Huang [61], [62], Chen, Huang and Lee [63] Dentscheva and Helbig [79], Göpfert, Tammer, Zalinescu [121], [122], Huang [135], Isac [139], Nemeth [202], Khanh [159], Tammer [253] have derived vector-valued variational principles for an objective function, which takes its values in general spaces. We remember that a Banach space Y is partially ordered, if in Y a reflexive, transitive and antisymmetrical relation is given. This is exactly realized, if in Y a convex, pointed cone C with 0 ∈ C exists, and this means, that exactly the elements of C dominate 0 ∈ Y. Therefore, given a nonempty set A in Y, a point z^{0} ∈ A is called a maximal point of A, if

{z^{0}} = A ∩ (z^{0} + C)

(where z^{0} + C means the set {z^{0} + k : k ∈ C}), which means that z^{0} is the only point being simultaneously in A as well as in z^{0} + C.

Using the cone

K_{ϵ}^{R} := \{(x, r) ∈ X × R : r + \sqrt{ϵ} || x ||≤ 0\},  (1.2.14)

where ϵ > 0 is a parameter and X is a Banach space, Phelps showed that Ekeland’s variational principle [89] is a direct consequence of a maximal point theorem (Phelps [221]). This maximal point theorem simply says, that under the given boundedness and closedness conditions for A in the partial ordering (reflexive, transitive, antisymmetrical) defined by K_{ϵ}^{R} (i.e. that exactly the elements of K_{ϵ}^{R} dominate 0 ∈ X × R) any point (ξ, β) of A is dominated by at least one maximal point (x^{o}, r^{o}) of A. It is also possible to interprete Phelps in the form, that (x^{o}, r^{o}) solve the multicriteria optimization problem

(x, r) ⟶ max s.t.(x, r) ∈ A,  (1.2.15)

where X ×R is partially ordered by K_{ϵ}^{R}, with other words, there is no element (x, r), (x, r) ≠ (x^{o}, r^{o}), which dominates (x^{o}, r^{o}) relative to the cone K_{ϵ}^{R} as in (1.2.14). To find an access to Vector Optimization instead of R we consider a more general space Y: Let X and Y be Banach spaces, Y partially ordered by a given cone C ⊂ Y, where C is convex, pointed and closed with int C ≠ ∅. Furthermore, choose k^{o} ∈ int C such that || k^{o} ||= 1. For any ϵ with 0 < ϵ < 1 we define

K_{ϵ} := \{(x, y) ∈ X × Y : y + \sqrt{ϵ}k^{o} || x ||∈ −C\}.  (1.2.16)

K_{ϵ} is clearly a cone, it is the hypograph of the mapping −\sqrt{ϵ}k^{o} || . ||: X ⟶ Y, recalling

hypo(−\sqrt{ϵ}k^{o} || . ||) := \{(x, y) : y ∈ −\sqrt{ϵ}k^{o} || x || −C\}.

K_{ϵ} is pointed and closed, because C has those properties. The triangle inequality gives, that K_{ϵ} is convex. Finally, (0, −k^{o}) ∈ K_{ϵ} and (0, y) ∈ K_{ϵ} ∀y ∈ −C. Taking Y = R, C = R_{+} and k^{o} = 1 (1.2.16) coincides with (1.2.14). Given ȳ ∈ −C, the second inclusion in (1.2.16) is fulfilled for a normbounded set in X, otherwise there is a sequence {x_{n}} with {|| x_{n} ||} ⟶ ∞, and so

− || x_{n} || k^{o} ∈ C + \frac{ȳ}{\sqrt{ϵ}} ∀n,

−k^{o} ∈ C + \frac{1}{|| x_{n} ||} \frac{ȳ}{\sqrt{ϵ}} ∀n

but −k^{o} ∉ C. Of course, conical support points are related to supporting points of convex sets, an overview can be found in Phelps [221]. Especially, if X is a Banach space, X^{∗} its continuous dual, then taking γ with 0 < γ < 1, x^{∗} ∈ X^{∗} with || x^{∗} ||= 1, the cone

K(x^{∗}; γ) := {x ∈ X : γ || x ||≤ x^{∗}(x)} ⊂ X  (1.2.17)

is closed, linefree, convex and int K(x^{∗}, γ) ≠ ∅. It is just the cone, which was used by Bishop and Phelps to prove density theorems for supporting points and functionals for convex sets (although the corresponding maximal point lemma for the supporting points doesn’t use convexity). The cone (1.2.17) is related to the cones in (1.2.14) and (1.2.16) in an interesting kind: For clarifying we consider the cone (γ, X as above in (1.2.17))

K_{γ}^{X} = {x ∈ X : x − γk^{o} || x ||∈ C}  (1.2.18)

where C is a convex, pointed cone in X, 0 ∈ C, C ≠ {0}, and k^{o} ∈ C\{0}, || k^{o} ||= 1. Then K_{γ}^{X} is clearly a cone, it is convex (because of the triangle inequality) and linefree. Obviously k^{o} ∈ K_{γ}^{X} and if k ∈ K_{γ}^{X}, it follows

k ∈ γk^{o} || k || +C ⊂ C,

that means K_{γ}^{X} ⊂ C, and if γ ⟶ 0, then K_{γ}^{X} approximates C, and if γ ⟶ 1 then K_{γ}^{X} approximates the jet, given by k^{o}. Now we obtain two results relative to K_{γ}^{X} in (1.2.18):

(i) If x̄ ∈ K_{γ}^{X}, it follows x̄ − γk^{o} || x̄ ||∈ C. Suppose x̄^{∗} ∈ C^{∗} \ {0} and x̄^{∗}(k^{o}) = 1, then it holds

x̄^{∗}(x̄ − γk^{o} || x̄ ||) ≥ 0

and therefore

x̄^{∗}(x̄) ≥ γ || x̄ ||,

that means, comparing with (1.2.17)

K_{γ}^{X} ⊂ K(x̄^{∗}, γ).  (1.2.19)

(ii) Using K_{γ}^{X} instead of K_{ϵ} in (1.2.16) leads to an other kind of maximal point theorems which generalize support properties of sets (see [118]). In K_{γ}^{X} linear functionals x^{∗} don’t appear, so contrary to results obtained with help of (1.2.17), corresponding conical supporting points (with help of (1.2.18)) do not only belong to the convex hull of A but even to A itself.

Phelps [221] had shown a maximal point theorem in a product space X × R. The following maximal point theorem in a product space X × Y, where X and Y are Banach spaces (presented by Göpfert and Tammer in [119] and ander weaker assumptions by Göpfert, Tammer and Zalinescu in [121] and [122]) says, that under certain conditions for a closed set A ⊂ X × Y and for a cone C ⊂ Y, any point of A is dominated by a maximal point, where we use the partial ordering defined by the cone (1.2.16):

K_{ϵ} = \{(x, y) ∈ X × Y : y + \sqrt{ϵ}k^{o} || x ||∈ −C\}.

Theorem 1.2.9 Assume that A is a closed subset of X × Y, where X and Y are Banach spaces. Further, suppose that C ⊂ Y is a pointed, closed, convex cone with nonempty interior and bounded base, and assume k^{o} ∈ int C, ȳ ∈ Y, {y ∈ Y, ∃x ∈ X with (x, y) ∈ A} ⊂ ȳ + C, ϵ > 0.

Then for any point (x, y) ∈ A there exists (x_{o}, y_{o}) ∈ A such that

(x_{o}, y_{o}) ∈ A ∩ (K_{ϵ} + (x, y))

and

{(x_{o}, y_{o})} = A ∩ (K_{ϵ} + (x_{o}, y_{o})).

The essential idea in the proof of Theorem 1.2.9 is the following: Consider a sequence {A_{n}}_{n∈N} of sets:

A_{n} := A ∩ (K_{ϵ} + (x_{n}, y_{n})).  (1.2.20)

Under the given assumptions the sets A_{n} are closed. Define the sequence {(x_{n}, y_{n})} inductively as follows:

(x_{1}, y_{1}) = (x, y).

When we have obtained (x_{1}, y_{1}), (x_{2}, y_{2}),...,(x_{n}, y_{n}), then we choose (x_{n+1}, y_{n+1}) ∈ A_{n} such that using a cone C̃ with (C \ {0}) ⊂ int C̃

∄(x, y) ∈ A_{n} with y ∈ y_{n+1} − \frac{1}{n + 1}k^{o} − (C̃ \ \{0\}).  (1.2.21)

It is possible to show that diam A_{n} ⟶ 0. Applying Cantor’s Intersection Theorem the assertions in the theorem follow.

A variational principle for a vector optimization problem is a direct consequence of our Theorem 1.2.9. This following Theorem is an assertion about the existence of an efficient solution of a slightly perturbed vector optimization problem in a certain neighbourhood of an approximately efficient element of the original vector optimization problem. Variational principles for vector optimization problems were presented by Loridan [185], Nemeth [202], Khanh [159], Tammer [253], Göpfert and Tammer [119], Göpfert, Tammer, Zalinescu [121], [122], Isac [139], Dentscheva and Helbig [79], Chen and Huang [61], [62], [63], Huang [135] and others. In the following X and Y are considered to be real Banach spaces, U is a nonempty closed subset of X and C ⊂ Y is a pointed, closed, convex cone with nonempty interior and bounded base, k^{0} ∈ int C. Now, we introduce a function f : U ⟶ Y and assume that f is bounded from below. A function f : U ⟶ Y is said to be bounded from below on U if there exists an element z ∈ Y with f[U] ⊂ z + C.

Theorem 1.2.10 Suppose that U is a closed subset of X. Assume that f : U ⟶ Y is bounded from below and the epigraph of f is closed. Then for any ϵ > 0 and any f(x^{o}) ∈ Eff(f[U], C_{ϵk^{o}}) there exists an element x_{ϵ} ∈ U with

1. f(x_{ϵ}) ∈ f(x^{o}) − \sqrt{ϵ} || x_{ϵ} − x^{o} || k^{o} − C,

2. || x_{ϵ} − x^{o} ||≤ \sqrt{ϵ},

3. f_{ϵk^{o}}(x_{ϵ}) ∈ Eff(f_{ϵk^{o}}[U], C), where f_{ϵk^{o}}(x) := f(x) + \sqrt{ϵ} || x − x_{ϵ} || k^{o}.

1.2.2 Lagrangean multipliers and saddle point assertions

Consider a convex vector minimization problem

(P) Eff(f[A], C_{Y}),

where f : M ⟶ Y, g : M ⟶ Z, Y and Z are normed spaces, C_{Z} and C_{Y} are closed convex pointed cones in Z, Y, respectively and

A := {x ∈ M | g(x) ∈ −C_{Z}}.

As in ordinary scalar optimization Lagrange multipliers can be used for different purposes as for duality, sensitivity or for numerical approaches (compare Amahroq and Taa [2], Clarke [68], El Abdouni and Thibault [91], Li and Wang [181], Miettinen, [196], Minami [197], Tanaka [264, 265, 266, 267], Thibault [271], Wang [276]). In the following we derive existence results for Lagrangean multipliers. These results extend well known theorems on Lagrange multipiers in nonlinear programming considerably.

Lemma 1.2.1 Let X be a linear space, M a convex subset of X, Y and Z normed spaces, C_{Z} and C_{Y} are closed convex pointed cones in Z, Y, respectively, and int C_{Y} ≠ ∅. Assume that the mappings f : M ⟶ Y, g : M ⟶ Z are C_{Y} -convex, C_{Z}-convex, respectively, for which the following regularity assumptions are fulfilled:

(A1) int {(y, z) ∈ Y × Z | ∃x ∈ M : y ∈ f(x) + C_{Y} and z ∈ g(x) + C_{Z}} ≠ ∅,

(A2) ∃y_{0} ∈ cl f[A] and f[A] ∩ (y_{0} − (C_{Y} \ {0})) = ∅.

Then there exist y_{0}^{∗} ∈ C_{Y^{∗}}, z_{0}^{∗} ∈ C_{Z^{∗}} with (y_{0}^{∗}, z_{0}^{∗}) ≠ (0, 0) and it holds

(y_{0}^{∗}, y_{0}) = inf{(y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) | x ∈ M}.

Proof: Consider the following sets:

A := {(y, z) ∈ Y × Z | ∃x ∈ M : y ∈ f(x) + C_{Y} , z ∈ g(x) + C_{Z}}

and

B := {(y, z) ∈ Y × Z | y ∈ y_{0} − C_{Y} , z ∈ −C_{Z}}.

In order to apply a separation theorem for convex sets we show that the assumptions of the separation theorem are fulfilled.

A is convex since we get for (y^{1}, z^{1}) ∈ A, (y^{2}, z^{2}) ∈ A, 0 ≤ λ ≤ 1 and corresponding elements x^{1}, x^{2} ∈ M:

λy^{1} + (1 − λ)y^{2} ∈ λf(x^{1}) + C_{Y} + (1 − λ)f(x^{2}) + C_{Y}
⊂ f(λx^{1} + (1 − λ)x^{2}) + C_{Y} + C_{Y} ⊂ f(λx^{1} + (1 − λ)x^{2}) + C_{Y} ,

since C_{Y} is a convex cone and f a C_{Y} - convex mapping. Together with

λz^{1} + (1 − λ)z^{2} ∈ λg(x^{1}) + (1 − λ)g(x^{2}) + C_{Z} ⊂ g(λx^{1} + (1 − λ)x^{2}) + C_{Z},

because C_{Z} is a convex cone and g a C_{Z} - convex mapping we can conclude

(λy^{1} + (1 − λ)y^{2}, λz^{1} + (1 − λ)z^{2}) ∈ A.

Moreover, B is convex regarding the convexity of C_{Y} and C_{Z}. Under the assumption (A1) it holds int A ≠ ∅. In order to show int A ∩ B = ∅ we suppose: ∃(y, z) ∈ int A ∩ B. This implies

∃x ∈ M with g(x) ∈ z − C_{Z} ⊂ −C_{Z} and f(x) ∈ y − C_{Y} ⊂ y_{0} − C_{Y}

such that we get y_{0} = y = f(x) because of the definition of y_{0} in (A3.2.2) and since C_{Y} is a pointed convex cone. Regarding (y, z) ∈ int A it follows that there are an ϵ > 0 and U_{ϵ}(y) ⊂ Y, V_{ϵ}(z) ⊂ Z with U_{ϵ}(y) × V_{ϵ}(z) ⊂ A, especially for k^{0} ∈ C_{Y} \ {0}, ||k^{0}|| = 1 we consider (y − \frac{ϵ}{2}k^{0}, z) ∈ A, i.e., for some x′ ∈ M it holds

y_{0} − \frac{ϵ}{2}k^{0} = y − \frac{ϵ}{2}k^{0} ∈ f(x′) + C_{Y}

and

g(x′) ∈ z − C_{Z} ⊂ −C_{Z}.

This means x′ ∈ A and f(x′) ∈ y_{0} − (C_{Y} \ {0}) in contradiction to the definition of y_{0} in (A2). Now, it is possible to apply a separation theorem for convex sets. This separation theorem implies the existence of (y_{0}^{∗}, z_{0}^{∗}) ∈ (Y^{∗} × Z^{∗}) \ {0} such that

(z_{0}^{∗}, z^{1}) + (y_{0}^{∗}, y^{1}) ≥ (z_{0}^{∗}, z^{2}) + (y_{0}^{∗}, y^{2}) ∀(y^{1}, z^{1}) ∈ A and ∀(y^{2}, z^{2}) ∈ B.

(1.2.22)

In the following we show that y_{0}^{∗} ∈ C_{Y^{∗}} and z_{0}^{∗} ∈ C_{Z^{∗}}. If we suppose y_{0}^{∗} ∉ C_{Y^{∗}}, i.e., (y_{0}^{∗}, ȳ) < 0 for an element ȳ ∈ C_{Y}, we get for y := −ȳ ∈ −C_{Y} regarding that C_{Y} is a cone

sup{(y_{0}^{∗}, ny) | n ∈ N} = sup{n(y_{0}^{∗}, y) | n ∈ N} = ∞

in contradiction to the separation property (1.2.22). Analogously we can show z_{0}^{∗} ∈ C_{Z^{∗}}. For all x ∈ M it holds (f(x), g(x)) ∈ A and with (y_{0}, 0) ∈ B we get

inf{(z_{0}^{∗}, g(x)) + (y_{0}^{∗}, f(x)) | x ∈ M} ≥ (y_{0}^{∗}, y_{0}).

Now, consider a sequence {x_{n}}_{n∈N} in A = {x ∈ M | g(x) ∈ −C_{Z}} with

lim_{n→∞} f(x_{n}) = y_{0}.

Then we get

inf{(z_{0}^{∗}, g(x)) + (y_{0}^{∗}, f(x)) | x ∈ M} ≤ inf{(z_{0}^{∗}, g(x)) + (y_{0}^{∗}, f(x)) | x ∈ A}

≤ inf{(y_{0}^{∗}, f(x)) | x ∈ A} ≤ lim_{n→∞}(y_{0}^{∗}, f(x_{n})) = (y_{0}^{∗}, y_{0})

such that the equation holds.

Lemma 1.2.2 Additionally to the assumptions of Lemma 1.2.1 we suppose

(A3) (Generalized Slater condition) There exists an element x^{1} ∈ M such that for all z^{∗} ∈ C_{Z^{∗}} \ {0} it holds: (z^{∗}, g(x^{1})) < 0.

(i) Then there exist elements y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} and z_{0}^{∗} ∈ C_{Z^{∗}} with

(y_{0}^{∗}, y_{0}) = inf{(y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) | x ∈ M}.

(ii) If x_{0} ∈ A and f(x_{0}) = y_{0}, i. e., f(x_{0}) ∈ Eff(f[A], C_{Y}), then x_{0} is also a minimal solution of (y_{0}^{∗}, f(.)) + (z_{0}^{∗}, g(.)) on M and it holds

(z_{0}^{∗}, g(x_{0})) = 0.

Proof: (i) From Lemma 1.2.1 we can conclude that there exist y_{0}^{∗} ∈ C_{Y^{∗}}, z_{0}^{∗} ∈ C_{Z^{∗}} with (y_{0}^{∗}, z_{0}^{∗}) ≠ 0 and

(y_{0}^{∗}, y_{0}) = inf{(y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) | x ∈ M}.  (1.2.23)

Under the assumption (A3) we suppose y_{0}^{∗} = 0. Then we get in (1.2.22) with z^{1} = g(x^{1}), z^{2} = 0

(z_{0}^{∗}, g(x^{1})) ≥ (z_{0}^{∗}, 0) = 0.  (1.2.24)

Regarding (y_{0}^{∗}, z_{0}^{∗}) ≠ 0 it holds z_{0}^{∗} ≠ 0 and now together with the assumption (A3) a contradiction:

0 > (z_{0}^{∗}, g(x^{1})) ≥ 0,

because of (1.2.24). (ii) If x_{0} ∈ A and f(x_{0}) = y_{0} ∈ Eff(f[A], C_{Y}) then (1.2.23) implies

(y_{0}^{∗}, y_{0}) ≤ (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x_{0})) = (y_{0}^{∗}, y_{0})

such that (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) = inf{(y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) | x ∈ M} and (z_{0}^{∗}, g(x_{0})) = 0.

Remark: Conversely, if x_{0} ∈ M is a minimal solution of the Lagrangean (y_{0}^{∗}, f(.)) + (z_{0}^{∗}, g(.)) with g(x_{0}) ∈ −C_{Z} and (z_{0}^{∗}, g(x_{0})) = 0, then

f(x_{0}) ∈ w − Eff(f[A], C_{Y})

follows without regularity assumption:

(y_{0}^{∗}, f(x_{0})) = (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) ≤ (y_{0}^{∗}, f(x))

for all x ∈ M with g(x) ∈ −C_{Z} and f(x_{0}) ∈ w − Eff(f[A], C_{Y}) regarding y_{0}^{∗} ∈ C_{Y^{∗}} \ {0}.

Theorem 1.2.11 Suppose that (A1), (A2) and (A3) are fulfilled. Assume x_{0} ∈ M. Then it holds:

(i) If f(x_{0}) ∈ Eff(f[A], C_{Y}) then there exist y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} and z_{0}^{∗} ∈ C_{Z^{∗}} such that the following saddle point assertion is fulfilled:

(y_{0}^{∗}, f(x_{0})) + (z^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x))

(1.2.25)

∀x ∈ M, ∀z^{∗} ∈ C_{Z^{∗}}.

(ii) Conversely, if there are y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} and (x_{0}, z_{0}^{∗}) ∈ M × C_{Z^{∗}} such that the saddle point assertion (1.2.25) is fulfilled for all x ∈ M and z^{∗} ∈ C_{Z^{∗}}, then f(x_{0}) ∈ w − Eff(f[A], C_{Y}).

Proof: (i): Assume f(x_{0}) ∈ Eff(f[A], C_{Y}) Using Lemma 1.2.2, (ii), we get that there exist y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} and z_{0}^{∗} ∈ C_{Z^{∗}} with

(y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) ∀ x ∈ M.

Furthermore, regarding −g(x_{0}) ∈ C_{Z} it follows, again with Lemma 1.2.2, (ii),

(z^{∗}, g(x_{0})) ≤ 0 = (z_{0}^{∗}, g(x_{0})) ∀z^{∗} ∈ C_{Z^{∗}}.

This yields

(y_{0}^{∗}, f(x_{0})) + (z^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ∀ z^{∗} ∈ C_{Z^{∗}}.

Then both inequalities are fulfilled.

(ii): Suppose y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} and assume that the saddle point assertion is fulfilled for (x_{0}, z_{0}^{∗}) ∈ M × C_{Z^{∗}}. Then the first inequality implies

(z^{∗}, g(x_{0})) ≤ (z_{0}^{∗}, g(x_{0})) ∀z^{∗} ∈ C_{Z^{∗}}

such that we get regarding that C_{Z^{∗}} is a convex cone

(z^{∗} + z_{0}^{∗}, g(x_{0})) ≤ (z_{0}^{∗}, g(x_{0})) ∀z^{∗} ∈ C_{Z^{∗}},
(z^{∗}, g(x_{0})) ≤ 0 ∀z^{∗} ∈ C_{Z^{∗}}

and g(x_{0}) ∈ −C_{Z}. This implies

0 ≥ (z_{0}^{∗}, g(x_{0})) ≥ (0, g(x_{0})) = 0

since 0 ∈ C_{Z^{∗}} and so (z_{0}^{∗}, g(x_{0})) = 0. Consider now x ∈ M with g(x) ≤ 0, then we conclude from the second inequality in the saddle point assertion

(y_{0}^{∗}, f(x_{0})) = (y_{0}^{∗}, f(x_{0})) + (z_{0}^{∗}, g(x_{0})) ≤ (y_{0}^{∗}, f(x)) + (z_{0}^{∗}, g(x)) ≤ (y_{0}^{∗}, f(x))

and

(y_{0}^{∗}, f(x_{0})) ≤ (y_{0}^{∗}, f(x)) ∀x ∈ A.

This means f(x_{0}) ∈ w − Eff(f[A], C_{Y}).

Remark: A point (x_{0}, z_{0}^{∗}) ∈ M × C_{Z^{∗}} satisfying the property (1.2.25) for an element y_{0}^{∗} ∈ C_{Y^{∗}} \ {0} is called a y_{0}^{∗}-saddle point of the Lagrangean

Φ(x, z^{∗}) := (y_{0}^{∗}, f(x)) + (z^{∗}, g(x)), x ∈ M, z^{∗} ∈ C_{Z^{∗}}.

The relation (1.2.25) can be described by

Φ(x_{0}, z_{0}^{∗}) ∈ Min({Φ(x, z_{0}^{∗}) : x ∈ M}, y_{0}^{∗}),
Φ(x_{0}, z_{0}^{∗}) ∈ Max({Φ(x_{0}, z^{∗}) : z^{∗} ∈ C_{Z^{∗}}}, y_{0}^{∗}).

Remark: Taking M = R_{+}^{2} ⊂ Y = R^{2}, C_{Y} = C_{Z} = R_{+}^{2}, Y = Z = R^{2}, f = I (identity), g(x) = −x ∀x ∈ M we have A = {x ∈ R_{+}^{2}} and all assumptions of Theorem 1.2.11 are satisfied. Then x_{0} = (0, 1)^{T}, y_{0}^{∗} = (1, 0)^{T}, z_{0}^{∗} = (0, 0)^{T} is a y_{0}^{∗}-saddle point of the Lagrangean Φ : Φ(x, z^{∗}) = (y_{0}^{∗}, f(x)) + (z^{∗}, g(x)), x ∈ M, z^{∗} ∈ C_{Z^{∗}} since 0 + (z^{∗}, −x_{0}) ≤ 0 ≤ (x)_{1}. x_{0} is only weakly efficient as proved in the theorem. So we cannot expect a symmetrical assertion of the kind ”saddle-point iff efficiency”.

1.2.3 Conclusions

The application of vector-valued variational principles of Ekeland’s type for the characterization of approximately efficient elements for special classes of multicriteria optimization problems seems to be a successful and prospective research direction. Furthermore, the study of set-valued variants of Ekeland’s variational principle including applications is a young and growing field.

1.3 Duality

Keywords: Conjugation, Lagrangean, axiomatic duality, dual problem.

It is an old idea to try to complement a given optimization problem (f(x) → min with minimal value I) by a dual problem (g(y) → max with supremal value S, S ≤ I), remember the dual variational principles of Dirichlet and Thompson (cf. Zeidler [306]) or e. g. the paper of K. O. Friedrichs [98] or simply the pair of dual programs in linear optimization. The reasons for the introduction of a useful dual problem are the following:

• The dual problem has (under additional conditions) the same optimal value as the given ”primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics.

• An approximate solution of the given minimization problem gives an estimation of the minimal value I from above, whereas an approximate solution of the dual problem is an estimation of I from below, so that one gets intervals, which contain I.

• Recalling Lagrange method, saddle points, equilibrium points of two person games, shadow prices in economics, perturbation methods or dual variational principles, it becomes clear, that optimal dual variables often have a special meaning for the given problem.

Of course, the just listed advantages require a skilfully chosen dual program. Nevertheless, the mentioned points are motivation enough, to look for dual problems in multicriteria optimization too. There are a lot of papers, which are dedicated to that aim, also a lot of survey papers (see Jahn [146], Luc [190]). There are different approaches to duality:

Conjugation: Schönfeld [237], Breckner [34], [35], Zowe [307], [308], Nehse [200], Rosinger [231], Tanino, Sawaragi [269], Brumelle [40], Kawasaki [156], [157], Gerstewitz and Göpfert [107], Sawaragi, Nakayama and Tanino [233], Luc [190], Zalinescu [301].

Lagrangean: Corley [71], [72], Bitran [22], Gerstewitz, Iwanow [106], Göpfert and Gerth [116], Nehse [200], Jahn [143], [148], [146], Iwanow and Nehse [141], Nakayama [198], [199], Sawaragi, Nakayama, Tanino [233], Luc [190].

Axiomatic Duality : Luc [190], Luc, Jahn [189].

We explain some ideas and give examples.

1.3.1 Duality without scalarization

Let Y be a linear topological space partially ordered by a convex pointed cone C, C^{∗} := {y^{∗} ∈ Y^{∗} | y^{∗}(c) ≥ 0 ∀ c ∈ C} the dual cone to C, P, D nonempty subsets of Y, and let us consider the multicriteria problems

(P) Eff_{min}(P, C),
(D) Eff_{max}(D, C) := Eff_{min}(D, −C).

We speak of a pair of weakly dual problems, if

P ∩ (D − (C \ {0})) = ∅.  (1.3.26)

Since C is pointed, this is equivalent to (P + (C \ {0})) ∩ (D − (C \ {0})) = ∅. (P) and (D) are called strongly dual, if (1.3.26) holds together with 0 ∈ cl(P − D) or equivalently (P + O) ∩ (D + O) ≠ ∅ for all open neighbourhoods O of zero in Y. So strong duality means, that P and D touch each other or with other words, P and D don’t overlap. Otherwise we speak of a pair of dual problems with a duality gap (in the forementioned scalar case at the beginning of this chapter that would mean I > S). Having a pair of strongly dual multicriterial programs their feasible elements respectively give estimations of the set of efficient elements of (P) ”from above and below” (with respect to C)

Lemma 1.3.3 Assume (P), (D) to be weakly dual. If there are z^{0} ∈ P, ζ^{0} ∈ D such that z^{0} = ζ^{0}, then z^{0} is minimal for (P), ζ^{0} maximal for (D) and (P), (D) are strongly dual.

Proof z^{0} not minimal means, that there is z^{1} ∈ P such that z^{1} ∈ z^{0} − (C \ {0}) = ζ^{0} −(C \{0}), which contradicts (1.3.26). ζ^{0} to be maximal follows similarly. From z^{0} = ζ^{0} follows, that (P) and (D) are strongly dual.

To construct dual programs one uses - similar to ordinary ”scalar” programming - Lagrange technique. We apply such methods, for other approaches compare Jahn [146] and Luc [190]. The question is, to apply scalarization methods or not. We present at first duality theorems without taking into account scalarization of the objective function.

Instead of (P) we consider the following multicriteria problem with side constraints

(P_{1}) Eff_{min}(f(A), C),  (1.3.27)

with

A := {x ∈ M, g(x) ∈ C_{V} },

where X is a linear space, M a non-empty set in X, (V, C_{V}) a partially ordered linear topological space, (Y, C) as above. We add to Y (V) an element ∞ not belonging to the space Y (V), obtaining thus the space Y^{•}: Y^{•} = Y ∪ {∞} (V^{•}: V^{•} = V ∪ {∞}). We consider f : X → Y^{•}, domf = M, g : X → V^{•}, dom g = M. So a vector optimization problem is given as usual. To deal with it, we suppose, that to C in Y we choose an open and convex set B ⊂_{≠} Y such that

0 ∈ bdB, clB + (C \ {0}) ⊂ B.  (1.3.28)

In particular, clB could be a convex cone, such that B contains C \ {0}, because the second inclusion in (1.3.28) is equivalent to C \ {0} ⊂ B. So if dim Y < ∞ such a cone B exists provided C is closed (and pointed). We introduce the following generalized Lagrangean, having another set M^{0∗},

L : M × M^{0∗} ⟶ Y ∪ {+∞_{Y} }

and assume

L(x, y) ∈ f(x) − cl B

for all x ∈ A and for all y ∈ M^{0∗}.

Now we are enabled to write down a problem (D_{1}) which can be considered as dual problem to (P_{1}) :

(D_{1}) Eff_{max}(f^{∗}(M^{∗}), C),  (1.3.29)

with

M^{∗} := {y ∈ M^{0∗} : Eff_{min}(L(M, y), B) ≠ ∅},

where B as explained above, and f^{∗}(y) ∈ Eff_{min}({L(x, y) : x ∈ M}, B) ≠ ∅ ⊂ Y.

(D_{1}) is a so-called Lagrange dual problem to (P_{1}) as is easily to see, if we reduce to the finite dimensional scalar case (Y, C) = (R^{1}, R_{+}^{1}), x ∈ R^{n}, (V, C_{V}) = (R^{m}, R_{+}^{m}), B = intR_{+}^{1}, and if we take y as linear mapping λ. Then instead of (P_{1}) we get

f(x) → min s.t. x ∈ A = {x ∈ M, g(x) ∈ R_{+}^{m}}.

So, with M^{∗} = {λ ∈ −R_{+}^{m} : ∃x_{0} ∈ M : (f + λ^{T} g)(x) ≥ (f + λ^{T} g)(x_{0}) ∀x ∈ M and f^{∗}(λ) = min_{x∈M} (f + λ^{T} g)(x) > −∞} (D_{1}) has the well-known max-min form:

max_{λ∈M^{∗}} f^{∗}(λ) = max_{λ∈M^{∗}} min_{x∈M}(f + λ^{T} g)(x).

Lemma 1.3.4 Weak duality (P_{1}) and (D_{1}) are weakly dual, i.e.,

f(A) ∩ (f^{∗}(M^{∗}) − (C \ {0})) = ∅.  (1.3.30)

A strong duality theorem holds too. Therefore we assume a condition (V1) :

(f + y ∘ g)(M) ⊂ Eff_{min}((f + y ∘ g)(M), B) + (B ∪ {0})

for all y ∈ M^{∗}. Since one can choose the wrapping set B (or cone B) very close to C \ {0}, (V 1) means, that (f + y ∘ g)(M) cannot have improperly efficient elements with respect to C. This is reflected by the formulation of the next theorem which takes into account only primarily efficient points with respect to a set B which fulfils 1.3.28 refering to C.

Theorem 1.3.12 Strong duality theorem For (P_{1}) and (D_{1}) as given above, let (V 1) be fulfilled and assume f(\overline{x}) ∈ Eff_{min}(f(A), B) to be in Eff_{min}(f(A), C). Then f(\overline{x}) ∈ Eff_{max}(f^{∗}(M^{∗}), C).

Sometimes results like Theorem 1.3.12 are called a strong direct duality theorem, because a primal optimal solution is shown to be dually optimal. The converse direction is also interesting. This leads to converse duality theorems. To get such a theorem for our pair of optimization problems, we state a condition (V2):

(V2) Every solution of (D_{1}) is to be dominated by a properly efficient solution of (P_{1}), that means with a set B as in (1.3.28), for all d̃∈ Eff_{max}(f^{∗}(M^{∗}), C) there is an f(x̃) ∈ Eff_{max}(A), B) such that f(x̃) ∈ d̃+ C.

Theorem 1.3.13 Converse duality theorem Assume that (V1) and (V2) hold and d̃ ∈ Eff_{max}(f^{∗}(M^{∗}), C). Then there are \overline{x} ∈ A, B′ ⊂ Y with clB + B′ ⊂ B such that d̃= f(\overline{x}) ∈ Eff_{min}(f(A), B′).

The duality in the last section is an abstract and nonscalarized Lagrangean formalism for very general optimization problems. In Section 1.3.3 we will present pairs of dual problems for special classes of vector optimization problems.

1.3.2 Duality by scalarization

Consider (P) in the following form

(P_{2}) Eff(f_{min}[A], C),

with

A = {x ∈ M, g(x) ∈ C_{V} } ≠ ∅,

where f : M → Y, ∅ ≠ M ⊂ X, g : M → V, C_{V} a nonempty set in V and X, Y, V t.v.s., Y Hausdorff, Y partially ordered by C closed and with non empty interior. Additionally, Y is to be directed with respect to k^{0} ∈ C\{0}, i.e., Y = ∪{C+λk_{0}, λ ∈ R}. Let S be the set of sublinear surjective continuous strictly monotone (with respect to C) decreasing functionals s : Y → R is nonempty. Again we introduce a generalized Lagrangean L : M × M^{∗} ⟶ Y ∪ {+∞_{Y} } and assume for s ∈ S

sup_{y∈M^{∗}} s(L(x, y)) = s(f(x)) if g(x) ∈ C_{V} .

Using the functionals s ∈ S we define a dual problem to (P_{2}) :

(D_{2}) Eff_{max}(A_{D}, C)  (1.3.31)

where

A_{D} = {h ∈ Y : ∃s ∈ S, ∃y ∈ M^{∗} with s(h) = inf{s(L(x, y)) : x ∈ M} > −∞}.

It is possible to prove for the pair (P_{2}), (D_{2}) weak, strong direct and strong converse duality:

Lemma 1.3.5 Weak duality The pair (P_{2}), (D_{2}) fulfils

f(A) ∩ (A_{D} − (C \ {0})) = ∅.

For properly efficient elements z_{0} = f(x_{0}), that means, there is an s_{0} ∈ S such that s_{0}(f(x_{0})) = inf{s_{0}(f(x)), x ∈ A} > −∞, a strong duality theorem can be proved:

Theorem 1.3.14 Strong (direct) duality If z_{0} = f(x_{0}) with x_{0} ∈ A is properly efficient for (P_{2}), then it is efficient for (D_{2}), i.e., z_{0} ∈ Eff_{max}(A_{D}, C).

Under some more conditions also a converse duality theorem holds, that is, having z_{0} dually efficient then there is x_{0} ∈ Asuch that f(x_{0}) is efficient for (P_{2}). In order to prove such a converse duality theorem we use a characterization of efficient elements h ∈ A_{D} by scalarization.

Lemma 1.3.6 h^{0} ∈ Eff_{max}(A_{D}, C) iff s_{h}(h) ≤ s_{h}(h^{0}) ∀ h ∈ A_{D} where s_{h} corresponds to h according to the definition of A_{D} in (D_{2}).

Additionally a strong converse duality statement holds.

Theorem 1.3.15 Strong converse duality Under the conditions given for the pair (P_{2}), (D_{2}) and additionally, if P := f(A)+C is closed and if for all s ∈ S such that inf{s(f(x)) : x ∈ A} > −∞ there is an x_{0} ∈ A with inf{s(f(x)) : x ∈ A} = s(f(x_{0})) then a dually efficient element is properly primal efficient.

1.3.3 Duality assertions for vector-valued approximation problems

For special classes of vector optimization problems it is possible to derive a usefull dual problem explicitly. Consider a vector-valued approximation problems of the following form:

(P_{3}) Eff(f_{min}[A], C),

with

A := {x ∈ C_{X} : Bx − b ∈ C_{V} },

where now X is a linear normed space, partially ordered by a cone C_{X}, V is a reflexive Banach space, C_{V} a cone in V, B ∈ L(X, V), Y = R^{p}, partially ordered by a cone C such that C + R_{+}^{p} ⊂ C, all cones closed, convex, pointed, and intC^{∗} ≠ ∅, and f may have the form

f(x) = f_{1}(x) + ( \begin{matrix}α_{1}∥A^{1}x − a^{1}∥ \\ ... \\ α_{p}∥A^{p}x − a^{p}∥ \end{matrix} )  (1.3.32)

with f_{1} ∈ L(X, R^{p}), α_{i} ≥ 0 real (i = 1, ..., p), and a^{i} ∈ U, a given real normed space, A^{i} ∈ L(X, U), i = 1, ..., p.

(P_{3}), although a special case of (P_{1}) or (P_{2}) contains itself important special cases:

(i) (P_{3}) is a semi-infinite linear problem, if α_{i} = 0 ∀ i = 1, ..., p.

(ii) f can be interpreted as Lipschitz perturbed linear problem.

(iii) f_{1} = 0 gives a multicriteria location problem.

(iv) Consider f_{1}(x) → min, x ∈ A_{0} = {x ∈ A ≠ ∅, A^{i}x = a^{i}, i = 1, ..., p} = ∅. Then (P_{3}) is a parameterized surrogate problem.

(v) The general multicriteria approximation problem or location problem (cf. Gerth and Pöhler [109], Jahn [146], Tammer, K. and Tammer, Chr. [261] and Wanka [277, 278, 279]).

In order to use the above ideas one introduces a suitable Lagrangean L_{λ^{0}} to (P_{3}) for a given λ^{0} ∈ int C^{∗}:

L_{λ^{0}}(x, Y, u^{∗}) = λ^{0} ((\begin{matrix}... \\ α_{i}Y^{i}(a^{i} − A^{i}x) \\ ... \end{matrix}) + (\begin{matrix}... \\ f_{1i}(x) \\ ... \end{matrix})) + u^{∗}(b − Bx),

where

x ∈ M = C_{X}

and

(Y, u^{∗}) ∈ M^{0∗} := {(Y, u^{∗}) : Y = (Y^{1}, ..., Y^{p}), Y^{i} ∈ L(U_{i}, R), u^{∗} ∈ L(U_{i}, R),
u^{∗} ∈ C_{V}^{∗} , α_{i}||Y^{i}||_{∗} ≤ α_{i}, i = 1, ..., p}.

From this setting and using Jahn’s descalarization result (cf. Jahn [146]) one gets a dual program (D_{3}) to (P_{3}) with (strong duality if b ≠ 0):

(D_{3}) : Eff_{max}(A_{D}, C),

where

A_{D} := \{d = ( \begin{matrix}... \\ α_{i}Y^{i}(a^{i}) \\ ... \end{matrix} ) + Zb for (Y, Z) ∈ M^{∗}\}

and

M^{∗} = {(Y, Z) : Y = (Y^{1}, ..., Y^{p}), Y^{i} ∈ L(U_{i}, R), i = 1, ..., p, Z ∈ L(V, R^{p}),

∃λ^{∗} ∈ intC^{∗} such that ∑_{i=1}^{p} λ_{i}^{∗}(−α_{i}A^{i∗}Y^{i} + f_{1i}) − (ZB)^{∗}λ^{∗} ∈ C_{X}^{∗},

Z^{∗}λ^{∗} ∈ C_{V}^{∗} , α_{i}∥Y^{i}∥_{∗} ≤ α_{i}, i = 1, ..., p}.

We add some simple scalar examples.

Example 1.3.5 Consider instead of (P_{3}) the following scalar optimization problem

(P_{4}) f_{1}(x) + ∑_{i=1}^{p} α_{i}∥A^{i}x − a^{i}∥ → min s.t. x ∈ C_{X}, Bx − b ∈ C_{V}

where X, V, C_{X}, C_{V} , A^{i}, a^{i}(i = 1, ..., p) as above, f_{1} ∈ L(X, R^{1}) and α_{i} > 0 ∀ i = 1, ..., p. Since the sum in (P_{4}) is really a norm, (P_{4}) is an example of (P_{3}). Then we get a dual problem (D_{4}) as a special case of (D_{3}) :

(D_{4}) ∑_{i=1}^{p} α_{i}y^{i}(a^{i}) + z(b) → max s.t. y = (y^{1}, ..., y^{p}) ∈ L(U, R)^{p}∥y^{i}∥_{∗} ≤ 1 ∀ i, z ∈ L(V, R), z ∈ C_{V}^{∗} ,

∑_{i=1}^{p}(−α_{i}A^{i∗}y^{i} + f_{1}) − B^{∗}z ∈ C_{X}^{∗}.

(D_{4}) follows immediately from (D_{3}) since ∥y∥_{∗} ≤ 1 in (D_{3}) means max_{i=1,...,p}∥y^{i}∥_{∗} ≤ 1 and so ∥y^{i}∥_{∗} ≤ 1 for all i), because the maximum norm is suitable as dual norm to a sum of norms.

Example 1.3.6 The ordinary scalar classic location problem is contained in (P_{4}) : X, α_{1}, ..., α_{p}, a_{1}, ..., a_{p} as above, a_{1}, ..., a_{p} fixed,

(P_{5}) ∑_{i=1}^{p} α_{i}∥x − a^{i}∥ → min_{x∈X} (Fermat’s problem),

(D_{5}) ∑_{i=1}^{p} α_{i}y^{i}(a^{i}) → max s.t. y^{i} ∈ X^{∗}, ∥y^{i}∥_{∗} ≤ 1 ∀ i, ∑_{1}^{p} α_{i}y^{i} = 0.

Also the following linear scalar approximation problem (P_{6}) together with (D_{6}) is a special case of the pair (P_{4})|(D_{4}) : Let A be a convex closed subset of X and a ∈ X fixed. Consider the problem ∥x − a∥ → min s.t. x ∈ A. Then (D_{4}) reduces to

y(a) → max_{y∈X^{∗},∥y∥_{∗}≤1}.

Taking as A a closed linear subspace M ⊂ X, we get (P_{6}) ∥x − a∥ → min s.t. x ∈ M

(D_{6}) y(a) → max s.t. y ∈ M^{⊥}, ∥y∥_{∗} ≤ 1

where M^{⊥} is the annihilator to M : M^{⊥} = {y ∈ X^{∗} : y(x) = 0 ∀ x ∈ M}.

Example 1.3.7 We consider the following vector-valued location problem

(P) : Eff_{min}(f[R^{2}], C),

with

f(x) := ( \begin{matrix}|| x − a^{1} ||_{max} \\ || x − a^{2} ||_{max} \\ ⋯ \\ || x − a^{p} ||_{max} \end{matrix} ) ,

where x, a^{i} ∈ R^{2} (i = 1, ..., p) and

|| x ||_{max}= max{| x_{1} |, | x_{2} |}.

For applications in town planning it is important that we can choose different norms in the formulation of (P_{7}). The decision which of the norms will be used depends on the course of the roads in the city or in the district.

In the following we study the problem (P_{7}) with X = R^{2}, C = R_{+}^{p}, where R_{+}^{p} denotes the usual ordering cone in the p-dimensional Euclidean space.

Using duality assertions we will present an algorithm for solving (P_{7}) (compare Chalmet, Francis and Kolen [54], Gerth (Tammer) and Pöhler [109]). In [109] we have derived the following dual problem for (P_{7}):

(D) : Eff_{max}(f^{∗}[B], C),

with

f^{∗}(y) = ( \begin{matrix}Y^{1}a^{1} \\ ⋯ \\ Y^{n}a^{n} \end{matrix} ) ,

where

B = {Y = (Y^{1}, ..., Y^{p}), Y^{i} ∈ L(R^{2}, R) : ∃λ ∈ int R_{+}^{p} with

∑_{i=1}^{n} λ_{i}^{∗}Y^{i} = 0, and || Y^{i} ||_{∗}≤ 1 (i = 1, ..., p)}.

Here || · ||_{∗} denotes the Lebesgue-norm. We can use the conditions ∑_{i=1}^{p} λ_{i}^{∗}Y^{i} = 0, and || Y^{i} ||_{∗}≤ 1 (i = 1, ..., p) in order to derive an algorithm (cf. [109]). Consider the following sets with respect to the given facilities a^{i} ∈ R^{2} (i = 1, ..., p) which are related to the structure of the subdifferential of the maximum - norm:

s_{1}(a^{i}) = {x ∈ R^{2} | a_{1}^{i} − x_{1} = a_{2}^{i} − x_{2} ≥ 0},
s_{2}(a^{i}) = {x ∈ R^{2} | a_{1}^{i} − x_{1} = a_{2}^{i} − x_{2} ≤ 0},
s_{3}(a^{i}) = {x ∈ R^{2} | a_{1}^{i} − x_{1} = x_{2} − a_{2}^{i} ≥ 0},
s_{4}(a^{i}) = {x ∈ R^{2} | a_{1}^{i} − x_{1} = x_{2} − a_{2}^{i} ≤ 0},
s_{5}(a^{i}) = {x ∈ R^{2} | a_{2}^{i} − x_{2} >| a_{1}^{i} − x_{1} |},
s_{6}(a^{i}) = {x ∈ R^{2} | x_{2} − a_{2}^{i} >| a_{1}^{i} − x_{1} |},
s_{7}(a^{i}) = {x ∈ R^{2} | a_{1}^{i} − x_{1} >| a_{2}^{i} − x_{2} |},
s_{8}(a^{i}) = {x ∈ R^{2} | x_{1} − a_{1}^{i} >| a_{2}^{i} − x_{2} |}.

Moreover, we introduce the sets

S_{r} := {x ∈ N | ∃i ∈ {1, ..., p} and x ∈ s_{r}(a^{i})}

(r = 5, 6, 7, 8), where N denotes the smallest level set of the dual norm to the maximum-norm (Lebesgue-norm) containing the points a^{i} (i = 1, ..., p). Now, we are able to describe the following algorithm for solving the vector-valued location problem (see Gerth and Pöhler [109]):

Eff(f(X), C) = {(cl S_{5}∩cl S_{6})∪[(N \S_{5})∩(N \S_{6})]}∩{(cl S_{7}∩cl S_{8})∪[(N \S_{7})∩(N \S_{8})]}.

[figure]

Figure : The set of efficient elements of the vector-valued location problem (P_{7}) with the maximum norm.

1.3.4 Conclusions

There are many papers dealing with duality concepts (conjugation, Lagrangean and axiomatic duality) on the base of scalarization as well as without scalarization. In order to derive algorithms or inclusions for solutions (bounds) it is important to have duality assertions for special classes of vector optimization problems. Especially, it is essential to find useful dual problems, corresponding duality assertions and their application for the construction of primal-dual algorithms in the case of not necessary convex multicriteria optimization problems. Very often a skillful interpretation of found dual solutions or Lagrange variables can be useful.

1.4 Vector Variational Inequalities and Vector Equilibria

Keywords: Existence results, Mathematical Economy, game theory.

1.4.1 Vector variational inequalities and vector equilibrium problems

It is well known that optimization and nonlinear analysis are two branches of modern mathematics much developed lately. The results obtained in these areas used certain kinds of differentiability (directional derivative, subgradient, generalized subgradients, etc.), certain generalizations of convexity, geometrical methods (cone of feasible directions, normal and tangent cones, etc.), game theory, fixed point theory, topological degree, etc.

In recent years equilibrium problems have come to play a central role in the development and unification of diverse areas of mathematics, economics and physical sciences. Thus various problems of practical interest in optimization, variational inequalities, complementarity, economics, Nash equilibria and engineering involve equilibrium in their description.

The vector variational inequalities have been widely developed in recent years, and various solutions have been characterized and computed. These were first introduced by Giannessi [111] and further developed by many authors in different areas.

1.4. VECTOR VARIATIONAL INEQUALITIES AND VECTOR EQUILIBRIA35

Recent topics attracting considerable attention are equilibrium problems for vectorvalued mappings. Inspired by the scalar case, such problems have received different developments depending on the kind of order space where these have been considered.

There are different approaches to establish the existence of solutions of equilibrium problems in the vector case. The first one directly used a generalization of the well known lemma of Knaster, Kuratowski and Mazurkiewicz. The second one, as proposed by Oettli, leads to deduce, in a straightforward way, existence results for vector equilibrium problems from the results for vector equilibrium problems from the results about scalar case. A key tool for the study of such problems is an appropriate gauge function. The published papers could be grouped in the following way:

• Theory of Vector Optimization, Vector Equilibrium Problems and Vector Variational Inequalities (Ansari [3], [4], [5], [6]; Ansari and Siddigi [7]; Ansari, Siddigi and Yao [8]; Bianchi, Hadjisvvas and Schaible [19]; Blum and Oettli [25]; [26]; Chen and Craven [57], [58]; Conway [69]; Fan [94]; Fu [99]; Konnov and Yao [163]; Lee and Kum [176]; Lee, G.M., Kim and Lee, B.S. [172], [179], [180]; [173]; Lee, G.M., Kim and Lee, B.S., Cho [174]; Lee and Kum [178]; Li, Yang and Chen [182]; Lin [182]; Lin, Yang, Yao [184]; Noor [207]; Oettli [209]; Siddiqi, Ansari, Ahmad [240]; Rapcsak [228]; Yang [286], [287], [288], [289]; [290]; Yang and Chen [291]; Yang and Goh [292], Yao [294], [295]; Yu and Yao [299]).

• Existence of Solutions for Generalized Vector Variational Inequalities and Complementarity Problems (Chang, Thompson and Yuan [54]; Chen [55]; Chen and Hou [60]; Chen and Yang [64], [65]; Danilidis and Hadjisavvas [76]; Isac and Yuan [140]; Kazmi [158]; Lai and Yao [170]; Yin and Xu [298]; Qun [226]).

• Vector Variational Inequalities and Vector Equilibrium Problems with Set-Valued Mappings (Ding and Tarafdar [83]; Fu [100]; Parida and Sen [210]; Siddigi, Ansari, Khan [241]; Song [244])

• Vector Variational Inequalities, Vector Optimization and Scalarization (Chen, G.-Y., Chen, G.M. [56]; Chen and Craven [58]; Giannessi, Mastroeni and Pellegrini [112]; Goh and Yang [114]; Lee, G.M., Kim, Lee, B.S., Yen [175]).

• Stability of the Solution Sets (Yen and Phuong [297]; and Yen [296]).

• Monotone Vector Variational Inequalities ( Chowdhury and Tan [67]; Hadjisavvas and Schaible [129], Ding and Tarafdar [83]; Yen and Lee [296]).

1.4.2 Conclusions

Vector variational inequalities, vector equilibrium problems and vector complementarity problems contain vector optimization or Lagrangean problems or game theoretical problems as special cases, so they become more and more important for the modelling of more complicated practical problems in Economy, Engineering and Biosciences. Many authors have proved existence theorems for solutions of such problems; in recent papers generalizations of the Vector Variational Inequalities are considered. Further research on these topics will concern:

• Refinements of the existence conditions for solutions of vector variational inequalities, vector equilibrium problems and vector complementarity problems;

• Solution methods; properties of solution sets as sensitivity, stability or a-priori-estimations;

• Practical applications, e.g. for the Mathematical Economy.

1.5 Multicriteria Fractional Programming

Keywords: Multicriteria fractional programming, Dinkelbach-transformed problem, dialogue procedure.

1.5.1 Approximate solutions in multicriteria fractional programming

Many aims in real decision problems can be expressed by a fractional objective function (cf. [234]) so that the field of fractional optimization is very actual.

Example: The problem in economics to minimize a cost functional f_{1}(x) = a^{T} x and to maximize profitableness f_{2}(x) = \frac{b^{T} x}{c^{T} x}, where x, a, b, c ∈ R^{n}, can be formulated as the problem to determine the set of efficient elements of a multicriteria fractional programming problem

Eff(f[X], R_{+}^{2}),

with f(x) = (f_{1}(x), f_{2}(x)) and X ⊆ R^{n}.

For the case of optimization problems with only one fractional objective function Dinkelbach [83] has proposed a parametric solution approach. This approach is based on the relation to a special parametric problem, which is described without the original ratios. However, it requires, additionally, the generation of that unknown parameter value, for which equivalence to our original problem holds. Many other authors have already published results to generalize Dinkelbach’s idea also for efficient and properly efficient solutions of vectorial optimization problems with m fractional objective functions (cf. Bector and Chandra [12], Kaul and Lyall [155], Tammer, K., Chr. Tammer and Ohlendorf [262] and Weir [283]). But some of those results are not entirely correct. The aim of this section is to extend the results of Dinkelbach and other authors to different sets of approximate efficient and properly efficient solutions of multicriterial fractional optimization problems, which were introduced by Tammer [253] as well as Dentcheva and Helbig [80]. As a by-product we obtain the corrected formulations of corresponding results for the exact solutions. The main part of this section is devoted to the mentioned relations between the (approximate) solutions of the original multicriterial fractional problem and the transformed one. Moreover, we discuss possibilities to solve the transformed problem by a three-level dialogue approach following ideas of the book [126].

Consider for C = R_{+}^{m} a vectorial fractional optimization problem Eff(f(X), C), i.e., the problem

(P_{f}) f(x) = \frac{g(x)}{h(x)} → v − min! subject to x ∈ X ⊆ R^{n},

where \frac{g(x)}{h(x)} := (\frac{g_{1}(x)}{h_{1}(x)}, ..., \frac{g_{m}(x)}{h_{m}(x)})′ and h_{i}(x) > 0 ∀ x ∈ X, i = 1, ..., m. We show, that (P_{f}) is closely related to a multiparametric vector optimization problem P(λ), which we call the corresponding Dinkelbach-transformed problem, namely

P(λ) : H(x, λ) → v − min! subject to x ∈ X,

where H_{i}(x, λ) = g_{i}(x) − λ_{i}h_{i}(x), i = 1, ..., m and λ ∈ R^{m} is a parameter which must be chosen in a suitable way.

The original result of Dinkelbach [83] from 1967 (and also the foregoing result of Jagannathan [142] from 1966 for linear fractional problems) concerns the case m=1 with only one objective function and says that a given point x̄ is optimal for (P_{f}) \underline{iff} it is optimal for P(λ̄) with λ̄ = \frac{g(x̄)}{h(x̄)}.

Corresponding results for the sets of efficient and properly efficient solutions, respectively, of both problems in the case m ≥ 1 were given by Bector and Chandra [12], Kaul and Lyall [155], Weir [283] and others. Note that the formulation as well as the proof of the corresponding Lemma 1 in [155] and Theorem 4 in [283] are not entirely correct in the given form. Above all, the authors disregarded the fact that in the case of proper efficiency it is essential to assume, additionally, that all ratios \frac{h_{i}}{h_{j}} are bounded below by positive bounds (and not only by zero).

Now, in the following two theorems we formulate the relations between the sets of approximate solutions of (P_{f}) and P(λ).

Theorem 1.5.16 Let k^{0} ∈ intR_{+}^{m}, ϵ ≥ 0 and x̄ ∈ X. Then we have f(x̄) ∈ Eff(f(X), C_{ϵk^{0}}) ⟺ H(x̄, λ̄) ∈ Eff(H(X, λ̄), C_{ϵk̄}) for

λ̄_{i} = \frac{g_{i}(x̄)}{h_{i}(x̄)} − ϵk_{i}^{0}  and  k̄_{i} = k_{i}^{0}h_{i}(x̄),  ∀i = 1, ..., m.  (1.5.33)

For the special case ϵ = 0 we get the already mentioned result of [12] and [155] in the corrected form (namely, including the essential condition (1.5.33) for ϵ = 0, which actually was used there in the proofs but had been forgotten in the formulation of the statement).

Corollary 1.5.3 f(x̄) ∈ Eff(f(X), C) ⟺ H(x̄, λ̄) ∈ Eff(H(X, λ̄), C) for λ̄ according to (1.5.33) with ϵ = 0.

We denote the set of approximately properly efficient elements regarding Definitions 1.1.2, (ii) and 1.1.3 (compare [262]) by p − Eff(f(X), C_{ϵk^{0}}).

Theorem 1.5.17 Let be k^{0} ∈ intR_{+}^{m}, ϵ ≥ 0 and x̄ ∈ X and assume that there is a positive number γ such that for all i,j= 1,..., m and all x ∈ X it holds \frac{h_{i}(x)}{h_{j}(x)} ≥ γ. Then we have f(x̄) ∈ p − Eff(f(X), C_{ϵk^{0}}) ⇔ H(x̄, λ̄) ∈ p − Eff(H(X, λ̄), C_{ϵk̄}) for λ̄ and k̄ according to (1.5.33).

Note that the assertion of Theorem 1.5.17 does not remain true if we only assume (as it was done in [155] and [283]) h_{i}(x) > 0 on X for i=1,...,m, since then inf\{\frac{h_{i}(x)}{h_{j}(x)}/x ∈ X} = 0 is not excluded. This can be seen by the following small example. Let be n = 1, m = 2, X = {x ∈ R/x ≥ 0}, g_{1}(x) = e^{−x}, h_{1} = 1, g_{2}(x) = x^{2} + x + 1, h_{2}(x) = x^{2} + 1. Then, for instance, x̄ = 0 yields a properly efficient element (in the sense of Geoffrion)

H(x̄, λ̄) = (0, 0)^{T} for P(λ̄) with λ̄ = (1, 1)^{T} but (\frac{g_{1}(x̄)}{h_{1}(x̄)}, \frac{g_{2}(x̄)}{h_{2}(x̄)})^{T} = (1, 1)^{T} is not properly efficient in the sense of Geoffrion for (P_{f}). Similar examples can also be constructed for the other direction of Theorem 1.5.17. Of course, if all functions h_{i} are equal or if there are positive lower and upper bounds for all functions h_{i} on X, the required boundedness of all ratios h_{i}/h_{j} by positive bounds is satisfied.

1.5.2 Possibilities for a Solution Approach

The reason of using models of vector optimization for solving concrete decision problems is the fact that very often it is impossible to formulate the interests of the decision maker a priori by only one objective function. As a natural consequence of such an incomplete knowledge about the underlying decision problem we can observe the phenomenon that in vector optimization we get a great number of ”solutions”, enjoying a priori the same rights. Of course, in practical decision problems the final aim must be to find such a feasible decision which corresponds to the decision maker’s interests in a certain ”optimal” way. As already described in [126] this can often be realized by organizing a learning process in form of a dialogue procedure in which one can compute and compare as much solutions as necessary to help the decision maker to express his individual interests more precisely. Such a dialogue procedure is usually a certain kind of a two-level algorithm and needs essentially a suitable parametric surrogate optimization problem related to the underlying vector optimization problem. Theoretically, all these ideas can also be applied directly to the fractional vector optimization problem (P_{f}) studied in the foregoing section. However, there may be computational difficulties to handle problems with complicated fractional objective functions. Moreover, there are also theoretical difficulties to ensure convexity properties of the surrogate problem which are to be solved in such a dialogue procedure. Note that even linear fractionals are not convex but only pseudoconvex and that for sums of fractionals even generalized convexity properties do not hold anymore. For this reason we want to discuss here possibilities to apply a dialogue procedure not directly to the original fractional problem (P_{f}) but to the corresponding Dinkelbach-transformed problem P(λ). However, in such an approach we have to overcome another difficulty, namely the generation of the essential parameter value λ̄ satisfying (1.5.33). Hence, different to dialogue procedures in the usual case, for our considered case of fractional vector optimization problems we propose a threelevel dialogue procedure. Let us explain our ideas for the mostly used set Eff(f(X), C) of efficient solutions ( ϵ = 0 ) of (P_{f}) and the mostly used surrogate problem, in which the artificial objective function is the weighted sum of the original objective functions. Applied to P(λ) our parametric surrogate problem has the form

P(λ, μ) :  F (x, λ, μ) → min!  subject to x ∈ X,

where μ > 0 and

F (x, λ, μ) = ∑_{i=1}^{m} μ_{i}(g_{i}(x) − λ_{i}h_{i}(x)).

The already mentioned three levels of a dialogue procedure for (P_{f}) may be characterized in the following way.

Level 1: Compare all stored results and decide to stop the procedure or not. If not, choose a new parameter vector μ.

Level 2: Find for the value of μ given from Level 1 a vector λ such that there exists a solution x of P(λ, μ) satisfying

H(x, λ) = 0.  (1.5.34)

Level 3: Find for the values μ and λ given in the Levels 1 and 2 a solution x of P(λ, μ) satisfying H(x, λ) = 0 and store x together with additional informations on x (especially the vector \frac{g(x)}{h(x)}). Go to Level 1.

Level 1 is the pure dialogue part in which we have to generate a new parameter value μ as long as we are not satisfied with the generated results. Level 3 can often be realized successfully by pathfollowing methods of parametric optimization. Because of the fact that possibilities to realize the Levels 1 and 3 are already described extensively in several papers (cf. [126]) we concentrate our considerations upon the second level. The typical difficulty in this level is the fact that the essential equation (1.5.34) is only given implicitly since the solution x of the third level is unknown at the time in which we have to solve the second level.

Let us study Level 2 under the following additional assumption A4). Here we take the symbol PC^{r}, (r ≥ 1) to denote the class of those C^{r−1}-functions for which the derivations of order r-1 are piecewise C^{1} (cf. [235], [263]).

(A) X = {x ∈ R^{n} / q_{l}(x) ≤ 0, l = 1, ..., p}, g_{i}, h_{i}, q_{l} ∈ PC^{2} and there exists a strongly stable stationary point (x^{∗}, u^{∗}) of P(λ^{∗}, μ^{∗}) satisfying condition (1.5.34).

Proposition 1.5.3 Let us assume (A). Then we have:

1. There are neighbourhoods U of λ^{∗}, V of μ^{∗} and W of (x^{∗}, u^{∗}) such that for each (λ, μ) ∈ U × V problem P(λ, μ) has a unique stationary point (x̄(λ, μ), ū(λ, μ)) in W.

2. The vector function (x̄, ū) belongs to the class PC^{1} on U.

3. The vector function G, defined on U by

G_{i}(λ) = g_{i}(x̄(λ, μ^{∗})) − λ_{i}h_{i}(x̄(λ, μ^{∗})), i = 1, ..., m

belongs to the class PC^{1}.

Obviously, under (A) condition (1.5.34) can be reformulated in the form

G(λ) = 0,  λ ∈ U,  (1.5.35)

To solve (1.5.34) we can apply suitable generalizations of the Newton-method for nonsmooth equations using generalized derivatives. In the papers [235] and [263] one can find possibilities to generate the generalized Jacobian of the vector functions x̄ and G. To ensure convergence to the (of course unknown) point λ^{∗} from assumption (A) usually one needs a suitable initial point λ^{0} in a sufficiently small neighbourhood of λ^{∗}. Among the great number of contributions concerning generalizations of the Newtonmethod to nonsmooth equations we refer here to the rather general results given in [168] and [227]. Usefull ideas to guarantee convergence in the second level even in the case that only approximate solutions of the third level may be generated (what may often be the case) can be found in [263].

Applying Lemma 2.1 in [152] the function φ̄(λ, μ) = F (x̄(λ, μ), λ, μ) belongs to the class PC^{2} and for each (λ, μ) ∈ U×V it holds \frac{∂}{∂λ_{i}} φ̄(λ, μ) = −μ_{i}h_{i}(x̄(λ, μ)) and, hence,

∑_{j=1}^{m} μ_{j}^{∗} \frac{∂}{∂λ_{i}} G_{j}(λ) = −μ_{i}^{∗}h_{i}(x̄(λ))  (1.5.36)

with x̄(λ) = x̄(λ, μ^{∗}). Hence, for the special case m=1 (in which we can put without loss of generality μ = 1) the function G belongs even to the class PC^{2} with ∇G(λ) = −h(x̄(λ)). In this way the iteration rule of the Newton-method has the very simple form

λ^{s+1} = λ^{s} − \frac{g(x̄(λ^{s})) − λ^{s}h(x̄(λ^{s}))}{−h(x̄(λ^{s}))} = f(x̄(λ^{s})),  (1.5.37)

which is nothing else than the iteration rule of Dinkelbach [83], who used this rule also under other assumptions as given in (A), since convergence results are very much easier to obtain in the one-dimensional case. Unfortunately, for m > 1 a formula of the type ∇G(λ) = −h(x̄(λ)) does not follow from (1.5.36).

1.5.3 Conclusions

Further research on multicriteria fractional programming includes:

• Duality assertions;

• Dependence of solutions or approximate solutions on variation of data of the problem;

• Solution procedures, convergence properties.

1.6 Multicriteria Control Problems

Keywords: ϵ-Pontrjagin’s minimum principle, structure of solutions.

1.6.1 Formulation of multicriteria Control Problems

In control theory often one has the problem to minimize more than one objective function, for instance a cost functional as well as the distance between the final state and a given point. To realize this task usually one takes as objective function a weighted sum of the different objectives. However, the more natural way would be to study the set of efficient points of a vector optimization problem with the given objective functions. It is well known that the weighted sum is only a special surrogate problem to find efficient points, which has the disadvantage that in the nonconvex case one cannot find all efficient elements in this way.

Necessary conditions for solutions of multiobjective dynamic programming or control problems were derived by several authors, see Klötzler [160], Benker and Kossert [17], Breckner [36], Gorochowik and Kirillowa [124], Gorochowik [123], [125] and Salukvadze [232]. It is difficult to show the existence of an optimal control (see Klötzler [162]), whereas suboptimal controls exist under very weak assumptions. So it is important to derive some assertions for suboptimal controls.

An ϵ-minimum principle in the sense of Pontrjagin for suboptimal controls of multicriteria control problems is derived by Tammer [260] applying a vector-valued variational principle.

Consider the system of differential equations

\begin{matrix}\frac{dx}{dt}(t) = & φ(t, x(t), u(t)), \\ x(0) = & x_{0} ∈ R^{n} \end{matrix} \}  (1.6.38)

and the control restriction

u(t) ∈ U,

which must hold almost everywhere on [0, T] with T > 0. We assume that

(C1) φ : [0, T] × R^{n} × U ⟶ R^{n} is continuous and U ⊂ R^{s} is a compact set.

The vector x(t) describes the state of the system, u(t) is the control at time t and belongs to the set U.

Furthermore, suppose that

(C2) \frac{∂φ}{∂x_{i}}, i=1,...,n, are continuous on [0, T] × R^{n} × U,

(C3) (x, φ(t, x, u)) ≤ c(1 + ||x||^{2}) for some c > 0.

Remark: Let be u : [0, T] ⟶ U a measurable control. Condition (C2) and the continuity of φ ensure that there exists a unique solution x of the differential equation (1.6.38) on [0, τ] for a sufficiently small τ > 0. By using Gronwall’s inequality condition (C3) implies

||x(t)||^{2} ≤ (||x_{0}||^{2} + 2cT)e^{2cT} ,

and, hence, ensures the existence of the solution on the whole time interval [0, T]. Moreover, the last inequality yields

||\frac{dx(t)}{dt}|| ≤ max\{φ(t, x, u) | (t, x, u) ∈ [0, T] × B^{0} × U\},

where B^{0} denotes the ball of radius (||x_{0}||^{2} + 2cT)^{\frac{1}{2}} e^{cT}. Applying Ascoli’s theorem, we see that the family of all trajectories x of the control system (1.6.38) is equicontinuous and bounded, and hence relatively compact in the uniform topology (compare Ekeland [89]).

In order to formulate the multicriteria control problem we introduce the objective function f : R^{n} ⟶ R^{m} and suppose that

(C4) f is a differentiable vector-valued function,

(C5) C ⊂ R^{m} is a pointed, closed convex cone with bounded base, k^{0} ∈ int C.

Now we formulate the multiobjective optimal control problem under the assumptions (C1) − (C5)

(P): Find some measurable control ū such that the corresponding trajectory x̄

satisfies

f(x(T)) ∉ f(x̄(T)) − (C \ {0})

for all solutions x of (1.6.38) (the pair (x̄, ū) is denoted as process corresponding to (1.6.38)).

1.6.2 An ϵ-minimum principle for multiobjective optimal control problem

It is well known that it is difficult to show the existence of optimal (or efficient) controls of (P), whereas suboptimal controls exist under very weak conditions. So it is important to derive some assertions for suboptimal controls. An application of a variational principle for vector optimization problems (see section 1.2, Theorem 1.2.10) yields an ϵ-minimum principle for (P), which is closely related to Pontryagin’s minimum principle (for ϵ = 0). We introduce the space V of controls, defined as the set of all measurable functions u : [0, T] ⟶ U with the metric

d(u_{1}, u_{2}) = measure{t ∈ [0, T] : u_{1}(t) ≠ u_{2}(t)}.

Then it holds Lemma 1.6.7 (Ekeland [89]) (V, d) is a complete metric space.

Lemma 1.6.8 (Ekeland [89]) The function F : u ⟶ f(x(T)) is continuous on V, where x(.) is the solution of (1.6.38) depending on u ∈ V.

Theorem 1.6.18 Consider the multicriteria control problem under the assumptions (C1) − (C5). For every ϵ > 0, there exists a measurable control u_{ϵ} with the corresponding admissible trajectory x_{ϵ} such that

1. f(x(T)) ∉ f(x_{ϵ}(T)) − ϵk^{0} − (C \ {0})

for all solutions x of (1.6.38),

2.

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ ( \begin{matrix}(φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} )

−ϵk^{0} − int C

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ}(.) = (p_{ϵ}^{1}(.), ..., p_{ϵ}^{m}(.)) is the solution of the linear differential system:

\begin{matrix}\frac{dp_{ϵi}^{s}}{dt} (t) = & − ∑_{j=1}^{n} \frac{∂φ_{j}}{∂x_{i}} (t, x_{ϵ}(t), u_{ϵ}(t))p_{ϵj}^{s}, \\ i = & 1, ..., n; \\ p_{ϵ}^{s}(T) = & f′_{s}(x_{ϵ}(T)), \\ s = & 1, ..., m. \end{matrix} \}  (1.6.39)

Remark : If we put ϵ = 0 then Theorem 1.6.18 coincides with the following assertion: Whenever there is a measurable control u_{ϵ} and the corresponding admissible trajectory x_{ϵ} with 1. f(x(T)) ∉ f(x_{ϵ}(T)) − (C \ {0})

for all solutions x of (1.6.38),

then

2.

( \begin{matrix}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) ∉ ( \begin{matrix}(φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{1}(t)) \\ ⋯ \\ (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}^{m}(t)) \end{matrix} ) − int C

for any u ∈ U and almost all t ∈ [0, T], where p_{ϵ}(.) is the solution of linear differential system (1.6.39) (the adjoint system).

This means that u_{ϵ} fulfills a minimum principle for multicriteria control problems in the sense of Pontrjagin (compare Gorochowik and Kirillowa [124] and Gorochowik [123], [125]).

Remark: Now let us study the special case Y = R^{1} and put ϵ = 0. Then Theorem 1.6.18 coincides with the following assertion: Whenever

(i) f(x_{ϵ}(T)) ≤ inf f(x(T)) holds,

then

(ii) (φ(t, x_{ϵ}(t), u_{ϵ}(t)) , p_{ϵ}(t)) ≤ min_{u∈U}(φ(t, x_{ϵ}(t), u(t)) , p_{ϵ}(t))

almost everywhere on [0,T], where p_{ϵ}(.) is the solution of (1.6.39).

This is the statement of Pontrjagin’s minimum principle. However, Theorem 1.6.18 holds even if optimal solutions do not exist.

1.6.3 Conclusions

There are only some papers dealing with vector-valued control problems. It is essential to find results, as for instance a minimum principle for approximate solutions, in order to have possibilities to analyse the structure of solutions or approximate ones. Furthermore, it is important to consider additionally non-deterministic aspects, that means multicriteria stochastic optimal control problems too.

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